Normal distribution














































































Normal Distribution

Probability density function

Probability density function for the normal distribution
The red curve is the standard normal distribution

Cumulative distribution function

Cumulative distribution function for the normal distribution
Notation N(μ2){displaystyle {mathcal {N}}(mu ,sigma ^{2})}{mathcal {N}}(mu ,sigma ^{2})
Parameters
μR{displaystyle mu in mathbb {R} }{displaystyle mu in mathbb {R} } = mean (location)
σ2>0{displaystyle sigma ^{2}>0}sigma ^{2}>0 = variance (squared scale)
Support x∈R{displaystyle xin mathbb {R} }{displaystyle xin mathbb {R} }
PDF 12πσ2e−(x−μ)22σ2{displaystyle {frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}{displaystyle {frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}
CDF 12[1+erf⁡(x−μσ2)]{displaystyle {frac {1}{2}}left[1+operatorname {erf} left({frac {x-mu }{sigma {sqrt {2}}}}right)right]}{displaystyle {frac {1}{2}}left[1+operatorname {erf} left({frac {x-mu }{sigma {sqrt {2}}}}right)right]}
Quantile μ2erf−1⁡(2F−1){displaystyle mu +sigma {sqrt {2}}operatorname {erf} ^{-1}(2F-1)}{displaystyle mu +sigma {sqrt {2}}operatorname {erf} ^{-1}(2F-1)}
Mean μ{displaystyle mu }mu
Median μ{displaystyle mu }mu
Mode μ{displaystyle mu }mu
Variance σ2{displaystyle sigma ^{2}}sigma ^{2}
Skewness 0{displaystyle 0}{displaystyle 0}
Ex. kurtosis 0{displaystyle 0}{displaystyle 0}
Entropy 12log⁡(2π2){displaystyle {frac {1}{2}}log(2pi esigma ^{2})}{displaystyle {frac {1}{2}}log(2pi esigma ^{2})}
MGF exp⁡t+σ2t2/2){displaystyle exp(mu t+sigma ^{2}t^{2}/2)}{displaystyle exp(mu t+sigma ^{2}t^{2}/2)}
CF exp⁡(iμt−σ2t2/2){displaystyle exp(imu t-sigma ^{2}t^{2}/2)}{displaystyle exp(imu t-sigma ^{2}t^{2}/2)}
Fisher information
I(μ)=(1/σ2002/σ2){displaystyle {mathcal {I}}(mu ,sigma )={begin{pmatrix}1/sigma ^{2}&0\0&2/sigma ^{2}end{pmatrix}}}{displaystyle {mathcal {I}}(mu ,sigma )={begin{pmatrix}1/sigma ^{2}&0\0&2/sigma ^{2}end{pmatrix}}}
I(μ2)=(1/σ2001/(2σ4)){displaystyle {mathcal {I}}(mu ,sigma ^{2})={begin{pmatrix}1/sigma ^{2}&0\0&1/(2sigma ^{4})end{pmatrix}}}{displaystyle {mathcal {I}}(mu ,sigma ^{2})={begin{pmatrix}1/sigma ^{2}&0\0&1/(2sigma ^{4})end{pmatrix}}}

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.


The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.


The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).


The probability density of the normal distribution is


f(x∣μ2)=12πσ2e−(x−μ)22σ2{displaystyle f(xmid mu ,sigma ^{2})={frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}{displaystyle f(xmid mu ,sigma ^{2})={frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}

where




  • μ{displaystyle mu }mu is the mean or expectation of the distribution (and also its median and mode),


  • σ{displaystyle sigma }sigma is the standard deviation, and


  • σ2{displaystyle sigma ^{2}}sigma ^{2} is the variance.




Contents






  • 1 Definition


    • 1.1 Standard normal distribution


    • 1.2 General normal distribution


    • 1.3 Notation


    • 1.4 Alternative parameterizations




  • 2 Properties


    • 2.1 Symmetries and derivatives


    • 2.2 Moments


    • 2.3 Fourier transform and characteristic function


    • 2.4 Moment and cumulant generating functions




  • 3 Cumulative distribution function


    • 3.1 Standard deviation and coverage


    • 3.2 Quantile function




  • 4 Zero-variance limit


  • 5 Central limit theorem


  • 6 Maximum entropy


  • 7 Operations on normal deviates


    • 7.1 Infinite divisibility and Cramér's theorem


    • 7.2 Bernstein's theorem




  • 8 Other properties


  • 9 Related distributions


    • 9.1 Operations on a single random variable


    • 9.2 Combination of two independent random variables


    • 9.3 Combination of two or more independent random variables


    • 9.4 Operations on the density function


    • 9.5 Extensions




  • 10 Normality tests


  • 11 Estimation of parameters


    • 11.1 Sample mean


    • 11.2 Sample variance


    • 11.3 Confidence intervals




  • 12 Bayesian analysis of the normal distribution


    • 12.1 Sum of two quadratics


      • 12.1.1 Scalar form


      • 12.1.2 Vector form




    • 12.2 Sum of differences from the mean


    • 12.3 With known variance


    • 12.4 With known mean


    • 12.5 With unknown mean and unknown variance




  • 13 Occurrence and applications


    • 13.1 Exact normality


    • 13.2 Approximate normality


    • 13.3 Assumed normality


    • 13.4 Produced normality




  • 14 Generating values from normal distribution


  • 15 Numerical approximations for the normal CDF


  • 16 History


    • 16.1 Development


    • 16.2 Naming




  • 17 See also


  • 18 Notes


  • 19 References


    • 19.1 Citations


    • 19.2 Sources




  • 20 External links





Definition



Standard normal distribution


The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ=0{displaystyle mu =0}mu =0 and σ=1{displaystyle sigma =1}sigma =1, and it is described by this probability density function:


φ(x)=12πe−12x2{displaystyle varphi (x)={frac {1}{sqrt {2pi }}}e^{-{frac {1}{2}}x^{2}}}{displaystyle varphi (x)={frac {1}{sqrt {2pi }}}e^{-{frac {1}{2}}x^{2}}}

The factor 1/2π{displaystyle 1/{sqrt {2pi }}}1/{sqrt {2pi }} in this expression ensures that the total area under the curve φ(x){displaystyle varphi (x)}varphi (x) is equal to one.[4] The factor 1/2{displaystyle 1/2}1/2 in the exponent ensures that the distribution has unit variance (i.e. the variance is equal to one), and therefore also unit standard deviation. This function is symmetric around x=0{displaystyle x=0}x=0, where it attains its maximum value 1/2π{displaystyle 1/{sqrt {2pi }}}1/{sqrt {2pi }} and has inflection points at x=+1{displaystyle x=+1}{displaystyle x=+1} and x=−1{displaystyle x=-1}x=-1.


Authors may differ also on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ2=1/2{displaystyle sigma ^{2}=1/2}{displaystyle sigma ^{2}=1/2}, that is


φ(x)=e−x2π{displaystyle varphi (x)={frac {e^{-x^{2}}}{sqrt {pi }}}}{displaystyle varphi (x)={frac {e^{-x^{2}}}{sqrt {pi }}}}

Stigler[5] goes even further, defining the standard normal with variance σ2=1/(2π){displaystyle sigma ^{2}=1/(2pi )}{displaystyle sigma ^{2}=1/(2pi )} :


φ(x)=e−πx2{displaystyle varphi (x)=e^{-pi x^{2}}}{displaystyle varphi (x)=e^{-pi x^{2}}}


General normal distribution


Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ{displaystyle sigma }sigma (the standard deviation) and then translated by μ{displaystyle mu }mu (the mean value):


f(x∣μ2)=1σφ(x−μσ).{displaystyle f(xmid mu ,sigma ^{2})={frac {1}{sigma }}varphi left({frac {x-mu }{sigma }}right).}{displaystyle f(xmid mu ,sigma ^{2})={frac {1}{sigma }}varphi left({frac {x-mu }{sigma }}right).}

The probability density must be scaled by 1/σ{displaystyle 1/sigma }1/sigma so that the integral is still 1.


If Z{displaystyle Z}Z is a standard normal deviate, then X=σZ+μ{displaystyle X=sigma Z+mu }{displaystyle X=sigma Z+mu } will have a normal distribution with expected value μ{displaystyle mu }mu and standard deviation σ{displaystyle sigma }sigma . Conversely, if X{displaystyle X}X is a normal deviate with parameters μ{displaystyle mu }mu and σ2{displaystyle sigma ^{2}}sigma ^{2}, then Z=(X−μ)/σ{displaystyle Z=(X-mu )/sigma }{displaystyle Z=(X-mu )/sigma } will have a standard normal distribution. This variate is called the standardized form of X{displaystyle X}X


Every normal distribution is the exponential of a quadratic function:


f(x)=eax2+bx+c{displaystyle f(x)=e^{ax^{2}+bx+c}}{displaystyle f(x)=e^{ax^{2}+bx+c}}

where a<0{displaystyle a<0}a<0 and c=b2/(4a)+ln⁡(−a/π)/2{displaystyle c=b^{2}/(4a)+ln(-a/pi )/2}{displaystyle c=b^{2}/(4a)+ln(-a/pi )/2}. In this form, the mean value is μ=−b/(2a){displaystyle mu =-b/(2a)}{displaystyle mu =-b/(2a)}, and the variance is σ2=−1/(2a){displaystyle sigma ^{2}=-1/(2a)}{displaystyle sigma ^{2}=-1/(2a)}. For the standard normal distribution, a=−1/2{displaystyle a=-1/2}{displaystyle a=-1/2}, b=0{displaystyle b=0}b=0, and c=−ln⁡(2π)/2{displaystyle c=-ln(2pi )/2}{displaystyle c=-ln(2pi )/2}.



Notation


The probability density of the standard Gaussian distribution (standard normal distribution) (with zero mean and unit variance) is often denoted with the Greek letter ϕ{displaystyle phi }phi (phi).[6] The alternative form of the Greek letter phi, φ{displaystyle varphi }varphi , is also used quite often.


The normal distribution is often referred to as N(μ2){displaystyle N(mu ,sigma ^{2})}N(mu ,sigma ^{2}) or N(μ2){displaystyle {mathcal {N}}(mu ,sigma ^{2})}{mathcal {N}}(mu ,sigma ^{2}).[7] Thus when a random variable X{displaystyle X}X is distributed normally with mean μ{displaystyle mu }mu and variance σ2{displaystyle sigma ^{2}}sigma ^{2}, one may write


X∼N(μ2).{displaystyle Xsim {mathcal {N}}(mu ,sigma ^{2}).}{displaystyle Xsim {mathcal {N}}(mu ,sigma ^{2}).}


Alternative parameterizations


Some authors advocate using the precision τ{displaystyle tau }tau as the parameter defining the width of the distribution, instead of the deviation σ{displaystyle sigma }sigma or the variance σ2{displaystyle sigma ^{2}}sigma ^{2}. The precision is normally defined as the reciprocal of the variance, 1/σ2{displaystyle 1/sigma ^{2}}{displaystyle 1/sigma ^{2}}.[8] The formula for the distribution then becomes


f(x)=τe−τ(x−μ)2/2.{displaystyle f(x)={sqrt {frac {tau }{2pi }}}e^{-tau (x-mu )^{2}/2}.}{displaystyle f(x)={sqrt {frac {tau }{2pi }}}e^{-tau (x-mu )^{2}/2}.}

This choice is claimed to have advantages in numerical computations when σ{displaystyle sigma }sigma is very close to zero and simplify formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.


Also the reciprocal of the standard deviation τ=1/σ{displaystyle tau ^{prime }=1/sigma }tau ^{prime }=1/sigma might be defined as the precision and the expression of the normal distribution becomes


f(x)=τe−)2(x−μ)2/2.{displaystyle f(x)={frac {tau ^{prime }}{sqrt {2pi }}}e^{-(tau ^{prime })^{2}(x-mu )^{2}/2}.}{displaystyle f(x)={frac {tau ^{prime }}{sqrt {2pi }}}e^{-(tau ^{prime })^{2}(x-mu )^{2}/2}.}

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution.



Properties


The normal distribution is the only absolutely continuous distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance.[9][10] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.[11][12]


The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.


The value of the normal distribution is practically zero when the value x{displaystyle x}x lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied.


The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the Cauchy distribution and the Lévy distribution.



Symmetries and derivatives


The normal distribution with density f(x){displaystyle f(x)}f(x) (mean μ{displaystyle mu }mu and standard deviation σ>0{displaystyle sigma >0}sigma >0) has the following properties:



  • It is symmetric around the point x=μ,{displaystyle x=mu ,}{displaystyle x=mu ,} which is at the same time the mode, the median and the mean of the distribution.[13]

  • It is unimodal: its first derivative is positive for x<μ,{displaystyle x<mu ,}{displaystyle x<mu ,} negative for x>μ,{displaystyle x>mu ,}{displaystyle x>mu ,} and zero only at x=μ.{displaystyle x=mu .}{displaystyle x=mu .}

  • The area under the curve and over the x{displaystyle x}x-axis is unity (i.e. equal to one).

  • Its density has two inflection points (where the second derivative of f{displaystyle f}f is zero and changes sign), located one standard deviation away from the mean, namely at x=μσ{displaystyle x=mu -sigma }{displaystyle x=mu -sigma } and x=μ.{displaystyle x=mu +sigma .}{displaystyle x=mu +sigma .}[13]

  • Its density is log-concave.[13]

  • Its density is infinitely differentiable, indeed supersmooth of order 2.[14]


Furthermore, the density φ{displaystyle varphi }varphi of the standard normal distribution (i.e. μ=0{displaystyle mu =0}mu =0 and σ=1{displaystyle sigma =1}{displaystyle sigma =1}) also has the following properties:



  • Its first derivative is φ(x)=−(x).{displaystyle varphi ^{prime }(x)=-xvarphi (x).}{displaystyle varphi ^{prime }(x)=-xvarphi (x).}

  • Its second derivative is φ(x)=(x2−1)φ(x){displaystyle varphi ^{prime prime }(x)=(x^{2}-1)varphi (x)}{displaystyle varphi ^{prime prime }(x)=(x^{2}-1)varphi (x)}

  • More generally, its nth derivative is φ(n)(x)=(−1)nHen⁡(x)φ(x),{displaystyle varphi ^{(n)}(x)=(-1)^{n}operatorname {He} _{n}(x)varphi (x),}{displaystyle varphi ^{(n)}(x)=(-1)^{n}operatorname {He} _{n}(x)varphi (x),} where Hen⁡(x){displaystyle operatorname {He} _{n}(x)}{displaystyle operatorname {He} _{n}(x)} is the nth (probabilist) Hermite polynomial.[15]

  • The probability that a normally distributed variable X{displaystyle X}X with known μ{displaystyle mu }mu and σ{displaystyle sigma }sigma is in a particular set, can be calculated by using the fact that the fraction Z=(X−μ)/σ{displaystyle Z=(X-mu )/sigma }{displaystyle Z=(X-mu )/sigma } has a standard normal distribution.



Moments



The plain and absolute moments of a variable X{displaystyle X}X are the expected values of Xp{displaystyle X^{p}}{displaystyle X^{p}} and |X|p{displaystyle |X|^{p}}{displaystyle |X|^{p}}, respectively. If the expected value μ{displaystyle mu }mu of X{displaystyle X}X is zero, these parameters are called central moments. Usually we are interested only in moments with integer order  p{displaystyle p} p.


If X{displaystyle X}X has a normal distribution, these moments exist and are finite for any p{displaystyle p}p whose real part is greater than −1. For any non-negative integer p{displaystyle p}p, the plain central moments are:[16]


E⁡[Xp]={0if p is odd,σp(p−1)!!if p is even.{displaystyle operatorname {E} left[X^{p}right]={begin{cases}0&{text{if }}p{text{ is odd,}}\sigma ^{p}(p-1)!!&{text{if }}p{text{ is even.}}end{cases}}}{displaystyle operatorname {E} left[X^{p}right]={begin{cases}0&{text{if }}p{text{ is odd,}}\sigma ^{p}(p-1)!!&{text{if }}p{text{ is even.}}end{cases}}}

Here n!!{displaystyle n!!}n!! denotes the double factorial, that is, the product of all numbers from n{displaystyle n}n to 1 that have the same parity as n.{displaystyle n.}n.


The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer p,{displaystyle p,}p,


E⁡[|X|p]=σp(p−1)!!⋅{2πif p is odd1if p is even}=σp⋅2p/2Γ(p+12)π{displaystyle operatorname {E} left[|X|^{p}right]=sigma ^{p}(p-1)!!cdot left.{begin{cases}{sqrt {frac {2}{pi }}}&{text{if }}p{text{ is odd}}\1&{text{if }}p{text{ is even}}end{cases}}right}=sigma ^{p}cdot {frac {2^{p/2}Gamma left({frac {p+1}{2}}right)}{sqrt {pi }}}}{displaystyle operatorname {E} left[|X|^{p}right]=sigma ^{p}(p-1)!!cdot left.{begin{cases}{sqrt {frac {2}{pi }}}&{text{if }}p{text{ is odd}}\1&{text{if }}p{text{ is even}}end{cases}}right}=sigma ^{p}cdot {frac {2^{p/2}Gamma left({frac {p+1}{2}}right)}{sqrt {pi }}}}

The last formula is valid also for any non-integer p>−1.{displaystyle p>-1.}{displaystyle p>-1.} When the mean μ0,{displaystyle mu neq 0,}{displaystyle mu neq 0,} the plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1F1{displaystyle {}_{1}F_{1}}{}_{1}F_{1} and U.{displaystyle U.}U.[citation needed]



E⁡[Xp]=σp⋅(−i2)pU(−p2,12,−12(μσ)2),{displaystyle operatorname {E} left[X^{p}right]=sigma ^{p}cdot (-i{sqrt {2}})^{p}Uleft(-{frac {p}{2}},{frac {1}{2}},-{frac {1}{2}}left({frac {mu }{sigma }}right)^{2}right),}{displaystyle operatorname {E} left[X^{p}right]=sigma ^{p}cdot (-i{sqrt {2}})^{p}Uleft(-{frac {p}{2}},{frac {1}{2}},-{frac {1}{2}}left({frac {mu }{sigma }}right)^{2}right),}

E⁡[|X|p]=σp⋅2p/2Γ(1+p2)π1F1(−p2,12,−12(μσ)2).{displaystyle operatorname {E} left[|X|^{p}right]=sigma ^{p}cdot 2^{p/2}{frac {Gamma left({frac {1+p}{2}}right)}{sqrt {pi }}}{}_{1}F_{1}left(-{frac {p}{2}},{frac {1}{2}},-{frac {1}{2}}left({frac {mu }{sigma }}right)^{2}right).}{displaystyle operatorname {E} left[|X|^{p}right]=sigma ^{p}cdot 2^{p/2}{frac {Gamma left({frac {1+p}{2}}right)}{sqrt {pi }}}{}_{1}F_{1}left(-{frac {p}{2}},{frac {1}{2}},-{frac {1}{2}}left({frac {mu }{sigma }}right)^{2}right).}


These expressions remain valid even if p{displaystyle p}p is not integer. See also generalized Hermite polynomials.
















































Order Non-central moment Central moment
1

μ{displaystyle mu }mu

0{displaystyle 0}{displaystyle 0}
2

μ2+σ2{displaystyle mu ^{2}+sigma ^{2}}{displaystyle mu ^{2}+sigma ^{2}}

σ2{displaystyle sigma ^{2}}sigma ^{2}
3

μ3+3μσ2{displaystyle mu ^{3}+3mu sigma ^{2}}{displaystyle mu ^{3}+3mu sigma ^{2}}

0{displaystyle 0}{displaystyle 0}
4

μ4+6μ2+3σ4{displaystyle mu ^{4}+6mu ^{2}sigma ^{2}+3sigma ^{4}}{displaystyle mu ^{4}+6mu ^{2}sigma ^{2}+3sigma ^{4}}

4{displaystyle 3sigma ^{4}}{displaystyle 3sigma ^{4}}
5

μ5+10μ2+15μσ4{displaystyle mu ^{5}+10mu ^{3}sigma ^{2}+15mu sigma ^{4}}{displaystyle mu ^{5}+10mu ^{3}sigma ^{2}+15mu sigma ^{4}}

0{displaystyle 0}{displaystyle 0}
6

μ6+15μ2+45μ4+15σ6{displaystyle mu ^{6}+15mu ^{4}sigma ^{2}+45mu ^{2}sigma ^{4}+15sigma ^{6}}{displaystyle mu ^{6}+15mu ^{4}sigma ^{2}+45mu ^{2}sigma ^{4}+15sigma ^{6}}

15σ6{displaystyle 15sigma ^{6}}{displaystyle 15sigma ^{6}}
7

μ7+21μ2+105μ4+105μσ6{displaystyle mu ^{7}+21mu ^{5}sigma ^{2}+105mu ^{3}sigma ^{4}+105mu sigma ^{6}}{displaystyle mu ^{7}+21mu ^{5}sigma ^{2}+105mu ^{3}sigma ^{4}+105mu sigma ^{6}}

0{displaystyle 0}{displaystyle 0}
8

μ8+28μ2+210μ4+420μ6+105σ8{displaystyle mu ^{8}+28mu ^{6}sigma ^{2}+210mu ^{4}sigma ^{4}+420mu ^{2}sigma ^{6}+105sigma ^{8}}{displaystyle mu ^{8}+28mu ^{6}sigma ^{2}+210mu ^{4}sigma ^{4}+420mu ^{2}sigma ^{6}+105sigma ^{8}}

105σ8{displaystyle 105sigma ^{8}}{displaystyle 105sigma ^{8}}

The expectation of X{displaystyle X}X conditioned on the event that X{displaystyle X}X lies in an interval [a,b]{displaystyle [a,b]}[a,b] is given by


E⁡[X∣a<X<b]=μσ2f(b)−f(a)F(b)−F(a){displaystyle operatorname {E} left[Xmid a<X<bright]=mu -sigma ^{2}{frac {f(b)-f(a)}{F(b)-F(a)}}}{displaystyle operatorname {E} left[Xmid a<X<bright]=mu -sigma ^{2}{frac {f(b)-f(a)}{F(b)-F(a)}}}

where f{displaystyle f}f and F{displaystyle F}F respectively are the density and the cumulative distribution function of X{displaystyle X}X. For b=∞{displaystyle b=infty }b=infty this is known as the inverse Mills ratio. Note that above, density f{displaystyle f}f of X{displaystyle X}X is used instead of standard normal density as in inverse Mills ratio, so here we have σ2{displaystyle sigma ^{2}}sigma ^{2} instead of σ{displaystyle sigma }sigma .



Fourier transform and characteristic function


The Fourier transform of a normal density f{displaystyle f}f with mean μ{displaystyle mu }mu and standard deviation σ{displaystyle sigma }sigma is[17]


f^(t)=∫f(x)e−itxdx=e−te−12(σt)2{displaystyle {hat {f}}(t)=int _{-infty }^{infty }f(x)e^{-itx},dx=e^{-imu t}e^{-{frac {1}{2}}(sigma t)^{2}}}{displaystyle {hat {f}}(t)=int _{-infty }^{infty }f(x)e^{-itx},dx=e^{-imu t}e^{-{frac {1}{2}}(sigma t)^{2}}}

where i{displaystyle i}i is the imaginary unit. If the mean μ=0{displaystyle mu =0}mu =0, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and standard deviation 1/σ{displaystyle 1/sigma }1/sigma . In particular, the standard normal distribution φ{displaystyle varphi }varphi is an eigenfunction of the Fourier transform.


In probability theory, the Fourier transform of the probability distribution of a real-valued random variable X{displaystyle X}X is closely connected to the characteristic function φX(t){displaystyle varphi _{X}(t)}varphi _{X}(t) of that variable, which is defined as the expected value of eitX{displaystyle e^{itX}}e^{{itX}}, as a function of the real variable t{displaystyle t}t (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable t{displaystyle t}t.[18] The relation between both is:


φX(t)=f^(−t){displaystyle varphi _{X}(t)={hat {f}}(-t)}{displaystyle varphi _{X}(t)={hat {f}}(-t)}


Moment and cumulant generating functions


The moment generating function of a real random variable X{displaystyle X}X is the expected value of etX{displaystyle e^{tX}}{displaystyle e^{tX}}, as a function of the real parameter t{displaystyle t}t. For a normal distribution with density f{displaystyle f}f, mean μ{displaystyle mu }mu and deviation σ{displaystyle sigma }sigma , the moment generating function exists and is equal to


M(t)=E⁡[etX]=f^(it)=eμte12σ2t2{displaystyle M(t)=operatorname {E} [e^{tX}]={hat {f}}(it)=e^{mu t}e^{{tfrac {1}{2}}sigma ^{2}t^{2}}}{displaystyle M(t)=operatorname {E} [e^{tX}]={hat {f}}(it)=e^{mu t}e^{{tfrac {1}{2}}sigma ^{2}t^{2}}}

The cumulant generating function is the logarithm of the moment generating function, namely


g(t)=ln⁡M(t)=μt+12σ2t2{displaystyle g(t)=ln M(t)=mu t+{tfrac {1}{2}}sigma ^{2}t^{2}}{displaystyle g(t)=ln M(t)=mu t+{tfrac {1}{2}}sigma ^{2}t^{2}}

Since this is a quadratic polynomial in t{displaystyle t}t, only the first two cumulants are nonzero, namely the mean μ{displaystyle mu }mu and the variance σ2{displaystyle sigma ^{2}}sigma ^{2}.



Cumulative distribution function


The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter Φ{displaystyle Phi }Phi (phi), is the integral


Φ(x)=12πxe−t2/2dt{displaystyle Phi (x)={frac {1}{sqrt {2pi }}}int _{-infty }^{x}e^{-t^{2}/2},dt}{displaystyle Phi (x)={frac {1}{sqrt {2pi }}}int _{-infty }^{x}e^{-t^{2}/2},dt}

The related error function erf⁡(x){displaystyle operatorname {erf} (x)}operatorname{erf}(x) gives the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range [−x,x]{displaystyle [-x,x]}[-x,x]; that is


erf⁡(x)=2π0xe−t2dt{displaystyle operatorname {erf} (x)={frac {2}{sqrt {pi }}}int _{0}^{x}e^{-t^{2}},dt}{displaystyle operatorname {erf} (x)={frac {2}{sqrt {pi }}}int _{0}^{x}e^{-t^{2}},dt}

These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below.


The two functions are closely related, namely


Φ(x)=12[1+erf⁡(x2)]{displaystyle Phi (x)={frac {1}{2}}left[1+operatorname {erf} left({frac {x}{sqrt {2}}}right)right]}{displaystyle Phi (x)={frac {1}{2}}left[1+operatorname {erf} left({frac {x}{sqrt {2}}}right)right]}

For a generic normal distribution with density f{displaystyle f}f, mean μ{displaystyle mu }mu and deviation σ{displaystyle sigma }sigma , the cumulative distribution function is


F(x)=Φ(x−μσ)=12[1+erf⁡(x−μσ2)]{displaystyle F(x)=Phi left({frac {x-mu }{sigma }}right)={frac {1}{2}}left[1+operatorname {erf} left({frac {x-mu }{sigma {sqrt {2}}}}right)right]}{displaystyle F(x)=Phi left({frac {x-mu }{sigma }}right)={frac {1}{2}}left[1+operatorname {erf} left({frac {x-mu }{sigma {sqrt {2}}}}right)right]}

The complement of the standard normal CDF, Q(x)=1−Φ(x){displaystyle Q(x)=1-Phi (x)}Q(x)=1-Phi (x), is often called the Q-function, especially in engineering texts.[19][20] It gives the probability that the value of a standard normal random variable X{displaystyle X}X will exceed x{displaystyle x}x: P(X>x){displaystyle P(X>x)}{displaystyle P(X>x)}. Other definitions of the Q{displaystyle Q}Q-function, all of which are simple transformations of Φ{displaystyle Phi }Phi , are also used occasionally.[21]


The graph of the standard normal CDF Φ{displaystyle Phi }Phi has 2-fold rotational symmetry around the point (0,1/2); that is, Φ(−x)=1−Φ(x){displaystyle Phi (-x)=1-Phi (x)}Phi (-x)=1-Phi (x). Its antiderivative (indefinite integral) is


Φ(x)dx=xΦ(x)+φ(x)+C.{displaystyle int Phi (x),dx=xPhi (x)+varphi (x)+C.}{displaystyle int Phi (x),dx=xPhi (x)+varphi (x)+C.}

The CDF of the standard normal distribution can be expanded by Integration by parts into a series:


Φ(x)=12+12πe−x2/2[x+x33+x53⋅5+⋯+x2n+1(2n+1)!!+⋯]{displaystyle Phi (x)={frac {1}{2}}+{frac {1}{sqrt {2pi }}}cdot e^{-x^{2}/2}left[x+{frac {x^{3}}{3}}+{frac {x^{5}}{3cdot 5}}+cdots +{frac {x^{2n+1}}{(2n+1)!!}}+cdots right]}{displaystyle Phi (x)={frac {1}{2}}+{frac {1}{sqrt {2pi }}}cdot e^{-x^{2}/2}left[x+{frac {x^{3}}{3}}+{frac {x^{5}}{3cdot 5}}+cdots +{frac {x^{2n+1}}{(2n+1)!!}}+cdots right]}

where !!{displaystyle !!}!! denotes the double factorial.


An asymptotic expansion of the CDF for large x can also be derived using integration by parts; see Error function#Asymptotic expansion.[22]



Standard deviation and coverage





For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.


About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.


More precisely, the probability that a normal deviate lies in the range between μ{displaystyle mu -nsigma }{displaystyle mu -nsigma } and μ+nσ{displaystyle mu +nsigma }{displaystyle mu +nsigma } is given by


F(μ+nσ)−F(μ)=Φ(n)−Φ(−n)=erf⁡(n2).{displaystyle F(mu +nsigma )-F(mu -nsigma )=Phi (n)-Phi (-n)=operatorname {erf} left({frac {n}{sqrt {2}}}right).}{displaystyle F(mu +nsigma )-F(mu -nsigma )=Phi (n)-Phi (-n)=operatorname {erf} left({frac {n}{sqrt {2}}}right).}

To 12 significant figures, the values for n=1,2,…,6{displaystyle n=1,2,ldots ,6}{displaystyle n=1,2,ldots ,6} are:[23]

















































n{displaystyle n}n p=F(μ+nσ)−F(μ){displaystyle p=F(mu +nsigma )-F(mu -nsigma )}{displaystyle p=F(mu +nsigma )-F(mu -nsigma )} i.e. 1−p{displaystyle {text{i.e. }}1-p}{displaystyle {text{i.e. }}1-p} or 1 in p{displaystyle {text{or }}1{text{ in }}p}{displaystyle {text{or }}1{text{ in }}p}
OEIS
1 6999682689492137000♠0.682689492137 6999317310507863000♠0.317310507863



7000300000000000000♠3
.15148718753


OEIS: A178647
2 6999954499736104000♠0.954499736104 6998455002638960000♠0.045500263896



7001210000000000000♠21
.9778945080


OEIS: A110894
3 6999997300203937000♠0.997300203937 6997269979606300000♠0.002699796063



7002370000000000000♠370
.398347345


OEIS: A270712
4 6999999936657516000♠0.999936657516 6995633424840000000♠0.000063342484



7004157870000000000♠15787
.1927673

5 6999999999426697000♠0.999999426697 6993573303000000000♠0.000000573303



7006174427700000000♠1744277
.89362

6 6999999999998027000♠0.999999998027 6991197300000000000♠0.000000001973



7008506797345000000♠506797345
.897



Quantile function



The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:


Φ1(p)=2erf−1⁡(2p−1),p∈(0,1).{displaystyle Phi ^{-1}(p)={sqrt {2}}operatorname {erf} ^{-1}(2p-1),quad pin (0,1).}{displaystyle Phi ^{-1}(p)={sqrt {2}}operatorname {erf} ^{-1}(2p-1),quad pin (0,1).}

For a normal random variable with mean μ{displaystyle mu }mu and variance σ2{displaystyle sigma ^{2}}sigma ^{2}, the quantile function is


F−1(p)=μΦ1(p)=μ2erf−1⁡(2p−1),p∈(0,1).{displaystyle F^{-1}(p)=mu +sigma Phi ^{-1}(p)=mu +sigma {sqrt {2}}operatorname {erf} ^{-1}(2p-1),quad pin (0,1).}{displaystyle F^{-1}(p)=mu +sigma Phi ^{-1}(p)=mu +sigma {sqrt {2}}operatorname {erf} ^{-1}(2p-1),quad pin (0,1).}

The quantile Φ1(p){displaystyle Phi ^{-1}(p)}Phi ^{{-1}}(p) of the standard normal distribution is commonly denoted as zp{displaystyle z_{p}}{displaystyle z_{p}}. These values are used in hypothesis testing, construction of confidence intervals and Q-Q plots. A normal random variable X{displaystyle X}X will exceed μ+zpσ{displaystyle mu +z_{p}sigma }{displaystyle mu +z_{p}sigma } with probability 1−p{displaystyle 1-p}1-p, and will lie outside the interval μ±zpσ{displaystyle mu pm z_{p}sigma }{displaystyle mu pm z_{p}sigma } with probability 2(1−p){displaystyle 2(1-p)}{displaystyle 2(1-p)}. In particular, the quantile z0.975{displaystyle z_{0.975}}{displaystyle z_{0.975}} is 1.96; therefore a normal random variable will lie outside the interval μ±1.96σ{displaystyle mu pm 1.96sigma }mu pm 1.96sigma in only 5% of cases.


The following table gives the quantile zp{displaystyle z_{p}}{displaystyle z_{p}} such that X{displaystyle X}X will lie in the range μ±zpσ{displaystyle mu pm z_{p}sigma }{displaystyle mu pm z_{p}sigma } with a specified probability p{displaystyle p}p. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:.[24][25] NOTE: the following table shows 2erf−1⁡(p)=Φ1(p+12){displaystyle {sqrt {2}}operatorname {erf} ^{-1}(p)=Phi ^{-1}left({frac {p+1}{2}}right)}{displaystyle {sqrt {2}}operatorname {erf} ^{-1}(p)=Phi ^{-1}left({frac {p+1}{2}}right)}, not Φ1(p){displaystyle Phi ^{-1}(p)}Phi ^{{-1}}(p) as defined above.




















































p{displaystyle p}p
zp{displaystyle z_{p}}{displaystyle z_{p}}
 
p{displaystyle p}p
zp{displaystyle z_{p}}{displaystyle z_{p}}
0.80 7000128155156554500♠1.281551565545 0.999
7000329052673149200♠3.290526731492
0.90 7000164485362695100♠1.644853626951 0.9999
7000389059188641300♠3.890591886413
0.95 7000195996398454000♠1.959963984540 0.99999
7000441717341346900♠4.417173413469
0.98 7000232634787404100♠2.326347874041 0.999999
7000489163847569899♠4.891638475699
0.99 7000257582930354900♠2.575829303549 0.9999999
7000532672388638400♠5.326723886384
0.995 7000280703376834400♠2.807033768344 0.99999999
7000573072886823600♠5.730728868236
0.998 7000309023230616800♠3.090232306168 0.999999999
7000610941020486900♠6.109410204869

For small p{displaystyle p}p, the quantile function has the useful asymptotic expansion
Φ1(p)=−ln⁡1p2−ln⁡ln⁡1p2−ln⁡(2π)+o(1).{displaystyle Phi ^{-1}(p)=-{sqrt {ln {frac {1}{p^{2}}}-ln ln {frac {1}{p^{2}}}-ln(2pi )}}+{mathcal {o}}(1).}{displaystyle Phi ^{-1}(p)=-{sqrt {ln {frac {1}{p^{2}}}-ln ln {frac {1}{p^{2}}}-ln(2pi )}}+{mathcal {o}}(1).}



Zero-variance limit


In the limit when σ{displaystyle sigma }sigma tends to zero, the probability density f(x){displaystyle f(x)}f(x) eventually tends to zero at any x≠μ{displaystyle xneq mu }{displaystyle xneq mu }, but grows without limit if x=μ{displaystyle x=mu }{displaystyle x=mu }, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when σ=0{displaystyle sigma =0}sigma =0.


However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function" δ{displaystyle delta }delta translated by the mean μ{displaystyle mu }mu , that is f(x)=δ(x−μ).{displaystyle f(x)=delta (x-mu ).}{displaystyle f(x)=delta (x-mu ).}
Its CDF is then the Heaviside step function translated by the mean μ{displaystyle mu }mu , namely


F(x)={0if x<μ1if x≥μ{displaystyle F(x)={begin{cases}0&{text{if }}x<mu \1&{text{if }}xgeq mu end{cases}}}{displaystyle F(x)={begin{cases}0&{text{if }}x<mu \1&{text{if }}xgeq mu end{cases}}}


Central limit theorem




As the number of discrete events increases, the function begins to resemble a normal distribution




Comparison of probability density functions, p(k){displaystyle p(k)}p(k) for the sum of n{displaystyle n}n fair 6-sided dice to show their convergence to a normal distribution with increasing na{displaystyle na}{displaystyle na}, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).



The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where X1,…,Xn{displaystyle X_{1},ldots ,X_{n}}{displaystyle X_{1},ldots ,X_{n}} are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance σ2{displaystyle sigma ^{2}}sigma ^{2} and Z{displaystyle Z}Z is their
mean scaled by n{displaystyle {sqrt {n}}}{sqrt {n}}


Z=n(1n∑i=1nXi){displaystyle Z={sqrt {n}}left({frac {1}{n}}sum _{i=1}^{n}X_{i}right)}Z={sqrt {n}}left({frac {1}{n}}sum _{i=1}^{n}X_{i}right)

Then, as n{displaystyle n}n increases, the probability distribution of Z{displaystyle Z}Z will tend to the normal distribution with zero mean and variance σ2{displaystyle sigma ^{2}}sigma ^{2}.


The theorem can be extended to variables (Xi){displaystyle (X_{i})}(X_{i}) that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.


Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.


The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:



  • The binomial distribution B(n,p){displaystyle B(n,p)}B(n,p) is approximately normal with mean np{displaystyle np}np and variance np(1−p){displaystyle np(1-p)}np(1-p) for large n{displaystyle n}n and for p{displaystyle p}p not too close to 0 or 1.

  • The Poisson distribution with parameter λ{displaystyle lambda }lambda is approximately normal with mean λ{displaystyle lambda }lambda and variance λ{displaystyle lambda }lambda , for large values of λ{displaystyle lambda }lambda .[26]

  • The chi-squared distribution χ2(k){displaystyle chi ^{2}(k)}{displaystyle chi ^{2}(k)} is approximately normal with mean k{displaystyle k}k and variance 2k{displaystyle 2k}2k, for large k{displaystyle k}k.

  • The Student's t-distribution t(ν){displaystyle t(nu )}{displaystyle t(nu )} is approximately normal with mean 0 and variance 1 when ν{displaystyle nu }nu is large.


Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.


A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.



Maximum entropy


Of all probability distributions over the reals with a specified mean μ{displaystyle mu }mu and variance σ2{displaystyle sigma ^{2}}sigma ^{2}, the normal distribution N(μ2){displaystyle N(mu ,sigma ^{2})}N(mu ,sigma ^{2}) is the one with maximum entropy.[27] If X{displaystyle X}X is a continuous random variable with probability density f(x){displaystyle f(x)}f(x), then the entropy of X{displaystyle X}X is defined as[28][29][30]


H(X)=−f(x)log⁡f(x)dx=12(1+log⁡(2σ)){displaystyle H(X)=-int _{-infty }^{infty }f(x)log f(x),dx={tfrac {1}{2}}(1+log(2sigma ^{2}pi ))}{displaystyle H(X)=-int _{-infty }^{infty }f(x)log f(x),dx={tfrac {1}{2}}(1+log(2sigma ^{2}pi ))}

where f(x)log⁡f(x){displaystyle f(x)log f(x)}{displaystyle f(x)log f(x)} is understood to be zero whenever f(x)=0{displaystyle f(x)=0}f(x)=0. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified variance, by using variational calculus. A function with two Lagrange multipliers is defined:


L=∫f(x)ln⁡(f(x))dx−λ0(1−f(x)dx)−λ2−f(x)(x−μ)2dx){displaystyle L=int _{-infty }^{infty }f(x)ln(f(x)),dx-lambda _{0}left(1-int _{-infty }^{infty }f(x),dxright)-lambda left(sigma ^{2}-int _{-infty }^{infty }f(x)(x-mu )^{2},dxright)}{displaystyle L=int _{-infty }^{infty }f(x)ln(f(x)),dx-lambda _{0}left(1-int _{-infty }^{infty }f(x),dxright)-lambda left(sigma ^{2}-int _{-infty }^{infty }f(x)(x-mu )^{2},dxright)}

where f(x){displaystyle f(x)}f(x) is, for now, regarded as some density function with mean μ{displaystyle mu }mu and standard deviation σ{displaystyle sigma }sigma .


At maximum entropy, a small variation δf(x){displaystyle delta f(x)}{displaystyle delta f(x)} about f(x){displaystyle f(x)}f(x) will produce a variation δL{displaystyle delta L}delta L about L{displaystyle L}L which is equal to 0:


0=δL=∫δf(x)(ln⁡(f(x))+1+λ0+λ(x−μ)2)dx{displaystyle 0=delta L=int _{-infty }^{infty }delta f(x)left(ln(f(x))+1+lambda _{0}+lambda (x-mu )^{2}right),dx}{displaystyle 0=delta L=int _{-infty }^{infty }delta f(x)left(ln(f(x))+1+lambda _{0}+lambda (x-mu )^{2}right),dx}

Since this must hold for any small δf(x){displaystyle delta f(x)}{displaystyle delta f(x)}, the term in brackets must be zero, and solving for f(x){displaystyle f(x)}f(x) yields:


f(x)=e−λ0−1−λ(x−μ)2{displaystyle f(x)=e^{-lambda _{0}-1-lambda (x-mu )^{2}}}f(x)=e^{-lambda _{0}-1-lambda (x-mu )^{2}}

Using the constraint equations to solve for λ0{displaystyle lambda _{0}}lambda _{0} and λ{displaystyle lambda }lambda yields the density of the normal distribution:


f(x,μ)=12πσ2e−(x−μ)22σ2{displaystyle f(x,mu ,sigma )={frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}{displaystyle f(x,mu ,sigma )={frac {1}{sqrt {2pi sigma ^{2}}}}e^{-{frac {(x-mu )^{2}}{2sigma ^{2}}}}}


Operations on normal deviates


The family of normal distributions is closed under linear transformations: if X is normally distributed with mean μ and standard deviation σ, then the variable Y = aX + b, for any real numbers a and b, is also normally distributed, with
mean + b and standard deviation |a|σ.


Also if X1 and X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their sum X1 + X2 will also be normally distributed,[proof] with mean μ1 + μ2 and variance σ12+σ22{displaystyle sigma _{1}^{2}+sigma _{2}^{2}}sigma _{1}^{2}+sigma _{2}^{2}.


In particular, if X and Y are independent normal deviates with zero mean and variance σ2, then X + Y and X − Y are also independent and normally distributed, with zero mean and variance 2σ2. This is a special case of the polarization identity.[31]


Also, if X1, X2 are two independent normal deviates with mean μ and deviation σ, and a, b are arbitrary real numbers, then the variable


X3=aX1+bX2−(a+b)μa2+b2+μ{displaystyle X_{3}={frac {aX_{1}+bX_{2}-(a+b)mu }{sqrt {a^{2}+b^{2}}}}+mu }X_{3}={frac {aX_{1}+bX_{2}-(a+b)mu }{sqrt {a^{2}+b^{2}}}}+mu

is also normally distributed with mean μ and deviation σ. It follows that the normal distribution is stable (with exponent α = 2).


More generally, any linear combination of independent normal deviates is a normal deviate.



Infinite divisibility and Cramér's theorem


For any positive integer n, any normal distribution with mean μ and variance σ2 is the distribution of the sum of n independent normal deviates, each with mean μ/n and variance σ2/n. This property is called infinite divisibility.[32]


Conversely, if X1 and X2 are independent random variables and their sum X1 + X2 has a normal distribution, then both X1 and X2 must be normal deviates.[33]


This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.[34]



Bernstein's theorem


Bernstein's theorem states that if X and Y are independent and X + Y and XY are also independent, then both X and Y must necessarily have normal distributions.[35][36]


More generally, if X1, …, Xn are independent random variables, then two distinct linear combinations ∑akXk and ∑bkXk will be independent if and only if all Xk's are normal and akbkσ 2
k
 
= 0
, where σ 2
k
 
denotes the variance of Xk.[35]



Other properties



  1. If the characteristic function φX of some random variable X is of the form φX(t) = eQ(t), where Q(t) is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that Q can be at most a quadratic polynomial, and therefore X is a normal random variable.[34] The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero cumulants.

  2. If X and Y are jointly normal and uncorrelated, then they are independent. The requirement that X and Y should be jointly normal is essential; without it the property does not hold.[37][38][proof] For non-normal random variables uncorrelatedness does not imply independence.

  3. The Kullback–Leibler divergence of one normal distribution X1N(μ1, σ21 )from another X2N(μ2, σ22 )is given by:[39]
    DKL(X1‖X2)=(μ1−μ2)22σ22+12(σ12σ22−1−ln⁡σ12σ22).{displaystyle D_{mathrm {KL} }(X_{1},|,X_{2})={frac {(mu _{1}-mu _{2})^{2}}{2sigma _{2}^{2}}}+{frac {1}{2}}left({frac {sigma _{1}^{2}}{sigma _{2}^{2}}}-1-ln {frac {sigma _{1}^{2}}{sigma _{2}^{2}}}right).}{displaystyle D_{mathrm {KL} }(X_{1},|,X_{2})={frac {(mu _{1}-mu _{2})^{2}}{2sigma _{2}^{2}}}+{frac {1}{2}}left({frac {sigma _{1}^{2}}{sigma _{2}^{2}}}-1-ln {frac {sigma _{1}^{2}}{sigma _{2}^{2}}}right).}

    The Hellinger distance between the same distributions is equal to


    H2(X1,X2)=1−12+σ22e−14(μ1−μ2)2σ12+σ22.{displaystyle H^{2}(X_{1},X_{2})=1-{sqrt {frac {2sigma _{1}sigma _{2}}{sigma _{1}^{2}+sigma _{2}^{2}}}}e^{-{frac {1}{4}}{frac {(mu _{1}-mu _{2})^{2}}{sigma _{1}^{2}+sigma _{2}^{2}}}}.}{displaystyle H^{2}(X_{1},X_{2})=1-{sqrt {frac {2sigma _{1}sigma _{2}}{sigma _{1}^{2}+sigma _{2}^{2}}}}e^{-{frac {1}{4}}{frac {(mu _{1}-mu _{2})^{2}}{sigma _{1}^{2}+sigma _{2}^{2}}}}.}


  4. The Fisher information matrix for a normal distribution is diagonal and takes the form
    I=(1σ20012σ4){displaystyle {mathcal {I}}={begin{pmatrix}{frac {1}{sigma ^{2}}}&0\0&{frac {1}{2sigma ^{4}}}end{pmatrix}}}{mathcal {I}}={begin{pmatrix}{frac {1}{sigma ^{2}}}&0\0&{frac {1}{2sigma ^{4}}}end{pmatrix}}


  5. Normal distributions belong to an exponential family with natural parameters θ1=μσ2{displaystyle textstyle theta _{1}={frac {mu }{sigma ^{2}}}}{displaystyle textstyle theta _{1}={frac {mu }{sigma ^{2}}}} and θ2=−12σ2{displaystyle textstyle theta _{2}={frac {-1}{2sigma ^{2}}}}{displaystyle textstyle theta _{2}={frac {-1}{2sigma ^{2}}}}, and natural statistics x and x2. The dual, expectation parameters for normal distribution are η1 = μ and η2 = μ2 + σ2.

  6. The conjugate prior of the mean of a normal distribution is another normal distribution.[40] Specifically, if x1, …, xn are iid N(μ, σ2) and the prior is μ ~ N(μ0, σ2
    0
    )
    , then the posterior distribution for the estimator of μ will be
    μx1,…,xn∼N(σ2nμ0+σ02x¯σ2n+σ02,(nσ2+1σ02)−1){displaystyle mu mid x_{1},ldots ,x_{n}sim {mathcal {N}}left({frac {{frac {sigma ^{2}}{n}}mu _{0}+sigma _{0}^{2}{bar {x}}}{{frac {sigma ^{2}}{n}}+sigma _{0}^{2}}},left({frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}right)^{-1}right)}{displaystyle mu mid x_{1},ldots ,x_{n}sim {mathcal {N}}left({frac {{frac {sigma ^{2}}{n}}mu _{0}+sigma _{0}^{2}{bar {x}}}{{frac {sigma ^{2}}{n}}+sigma _{0}^{2}}},left({frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}right)^{-1}right)}


  7. The family of normal distributions forms a manifold with constant curvature −1. The same family is flat with respect to the (±1)-connections ∇(e) and ∇(m).[41]



Related distributions



Operations on a single random variable


If X is distributed normally with mean μ and variance σ2, then



  • The exponential of X is distributed log-normally: eX ~ ln(N (μ, σ2)).

  • The absolute value of X has folded normal distribution: |X| ~ Nf (μ, σ2). If μ = 0 this is known as the half-normal distribution.

  • The absolute value of normalized residuals, |Xμ|/σ, has chi distribution with one degree of freedom: |Xμ|/σ ~ χ1(|Xμ|/σ).

  • The square of X/σ has the noncentral chi-squared distribution with one degree of freedom: X2/σ2 ~ χ21(μ2/σ2). If μ = 0, the distribution is called simply chi-squared.

  • The distribution of the variable X restricted to an interval [a, b] is called the truncated normal distribution.

  • (Xμ)−2 has a Lévy distribution with location 0 and scale σ−2.



Combination of two independent random variables


If X1 and X2 are two independent standard normal random variables with mean 0 and variance 1, then



  • Their sum and difference is distributed normally with mean zero and variance two: X1 ± X2N(0, 2).

  • Their product Z = X1·X2 follows the "product-normal" distribution[42] with density function fZ(z) = π−1K0(|z|), where K0 is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at z = 0, and has the characteristic function φZ(t) = (1 + t 2)−1/2.

  • Their ratio follows the standard Cauchy distribution: X1 / X2 ∼ Cauchy(0, 1).

  • Their Euclidean norm X12+X22{displaystyle {sqrt {X_{1}^{2}+X_{2}^{2}}}}{displaystyle {sqrt {X_{1}^{2}+X_{2}^{2}}}} has the Rayleigh distribution.



Combination of two or more independent random variables


  • If X1, X2, …, Xn are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with n degrees of freedom

X12+⋯+Xn2∼χn2.{displaystyle X_{1}^{2}+cdots +X_{n}^{2}sim chi _{n}^{2}.}{displaystyle X_{1}^{2}+cdots +X_{n}^{2}sim chi _{n}^{2}.}

  • If X1, X2, …, Xn are independent normally distributed random variables with means μ and variances σ2, then their sample mean is independent from the sample standard deviation,[43] which can be demonstrated using Basu's theorem or Cochran's theorem.[44] The ratio of these two quantities will have the Student's t-distribution with n − 1 degrees of freedom:

t=X¯μS/n=1n(X1+⋯+Xn)−μ1n(n−1)[(X1−)2+⋯+(Xn−)2]∼tn−1.{displaystyle t={frac {{overline {X}}-mu }{S/{sqrt {n}}}}={frac {{frac {1}{n}}(X_{1}+cdots +X_{n})-mu }{sqrt {{frac {1}{n(n-1)}}left[(X_{1}-{overline {X}})^{2}+cdots +(X_{n}-{overline {X}})^{2}right]}}}sim t_{n-1}.}{displaystyle t={frac {{overline {X}}-mu }{S/{sqrt {n}}}}={frac {{frac {1}{n}}(X_{1}+cdots +X_{n})-mu }{sqrt {{frac {1}{n(n-1)}}left[(X_{1}-{overline {X}})^{2}+cdots +(X_{n}-{overline {X}})^{2}right]}}}sim t_{n-1}.}

  • ' If X1, …, Xn, Y1, …, Ym are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom:[45]

F=(X12+X22+⋯+Xn2)/n(Y12+Y22+⋯+Ym2)/m∼Fn,m.{displaystyle F={frac {left(X_{1}^{2}+X_{2}^{2}+cdots +X_{n}^{2}right)/n}{left(Y_{1}^{2}+Y_{2}^{2}+cdots +Y_{m}^{2}right)/m}}sim F_{n,m}.}{displaystyle F={frac {left(X_{1}^{2}+X_{2}^{2}+cdots +X_{n}^{2}right)/n}{left(Y_{1}^{2}+Y_{2}^{2}+cdots +Y_{m}^{2}right)/m}}sim F_{n,m}.}


Operations on the density function


The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.



Extensions


The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.



  • The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector XRk is multivariate-normally distributed if any linear combination of its components k
    j=1
    aj Xj
    has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.


  • Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0


  • Complex normal distribution deals with the complex normal vectors. A complex vector XCk is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.


  • Matrix normal distribution describes the case of normally distributed matrices.


  • Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element hH is said to be normal if for any constant aH the scalar product (a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:


    • Brownian motion,


    • Brownian bridge,


    • Ornstein–Uhlenbeck process.




  • Gaussian q-distribution is an abstract mathematical construction that represents a "q-analogue" of the normal distribution.

  • the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. Note that this distribution is different from the Gaussian q-distribution above.


A random variable X has a two-piece normal distribution if it has a distribution



fX(x)=N(μ12) if x≤μ{displaystyle f_{X}(x)=N(mu ,sigma _{1}^{2}){text{ if }}xleq mu }{displaystyle f_{X}(x)=N(mu ,sigma _{1}^{2}){text{ if }}xleq mu }

fX(x)=N(μ22) if x≥μ{displaystyle f_{X}(x)=N(mu ,sigma _{2}^{2}){text{ if }}xgeq mu }{displaystyle f_{X}(x)=N(mu ,sigma _{2}^{2}){text{ if }}xgeq mu }


where μ is the mean and σ1 and σ2 are the standard deviations of the distribution to the left and right of the mean respectively.


The mean, variance and third central moment of this distribution have been determined[46]



E⁡(X)=μ+2π2−σ1){displaystyle operatorname {E} (X)=mu +{sqrt {frac {2}{pi }}}(sigma _{2}-sigma _{1})}{displaystyle operatorname {E} (X)=mu +{sqrt {frac {2}{pi }}}(sigma _{2}-sigma _{1})}

V⁡(X)=(1−)(σ2−σ1)2+σ2{displaystyle operatorname {V} (X)=left(1-{frac {2}{pi }}right)(sigma _{2}-sigma _{1})^{2}+sigma _{1}sigma _{2}}{displaystyle operatorname {V} (X)=left(1-{frac {2}{pi }}right)(sigma _{2}-sigma _{1})^{2}+sigma _{1}sigma _{2}}

T⁡(X)=2π2−σ1)[(4π1)(σ2−σ1)2+σ2]{displaystyle operatorname {T} (X)={sqrt {frac {2}{pi }}}(sigma _{2}-sigma _{1})left[left({frac {4}{pi }}-1right)(sigma _{2}-sigma _{1})^{2}+sigma _{1}sigma _{2}right]}{displaystyle operatorname {T} (X)={sqrt {frac {2}{pi }}}(sigma _{2}-sigma _{1})left[left({frac {4}{pi }}-1right)(sigma _{2}-sigma _{1})^{2}+sigma _{1}sigma _{2}right]}


where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.


One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:




  • Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.

  • The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.



Normality tests



Normality tests assess the likelihood that the given data set {x1, …, xn} comes from a normal distribution. Typically the null hypothesis H0 is that the observations are distributed normally with unspecified mean μ and variance σ2, versus the alternative Ha that the distribution is arbitrary. Many tests (over 40) have been devised for this problem, the more prominent of them are outlined below:




  • "Visual" tests are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.


    • Q-Q plot— is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it's a plot of point of the form (Φ−1(pk), x(k)), where plotting points pk are equal to pk = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.


    • P-P plot— similar to the Q-Q plot, but used much less frequently. This method consists of plotting the points (Φ(z(k)), pk), where z(k)=(x(k)−μ^)/σ^{displaystyle textstyle z_{(k)}=(x_{(k)}-{hat {mu }})/{hat {sigma }}}{displaystyle textstyle z_{(k)}=(x_{(k)}-{hat {mu }})/{hat {sigma }}}. For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).


    • Shapiro-Wilk test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.


    • Normal probability plot (rankit plot)




  • Moment tests:

    • D'Agostino's K-squared test

    • Jarque–Bera test




  • Empirical distribution function tests:


    • Lilliefors test (an adaptation of the Kolmogorov–Smirnov test)

    • Anderson–Darling test





Estimation of parameters



It is often the case that we don't know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x1, …, xn) from a normal N(μ, σ2) population we would like to learn the approximate values of parameters μ and σ2. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:


ln⁡L(μ2)=∑i=1nln⁡f(xi∣μ2)=−n2ln⁡(2π)−n2ln⁡σ2−12σ2∑i=1n(xi−μ)2.{displaystyle ln {mathcal {L}}(mu ,sigma ^{2})=sum _{i=1}^{n}ln f(x_{i}mid mu ,sigma ^{2})=-{frac {n}{2}}ln(2pi )-{frac {n}{2}}ln sigma ^{2}-{frac {1}{2sigma ^{2}}}sum _{i=1}^{n}(x_{i}-mu )^{2}.}{displaystyle ln {mathcal {L}}(mu ,sigma ^{2})=sum _{i=1}^{n}ln f(x_{i}mid mu ,sigma ^{2})=-{frac {n}{2}}ln(2pi )-{frac {n}{2}}ln sigma ^{2}-{frac {1}{2sigma ^{2}}}sum _{i=1}^{n}(x_{i}-mu )^{2}.}

Taking derivatives with respect to μ and σ2 and solving the resulting system of first order conditions yields the maximum likelihood estimates:


μ^=x¯1n∑i=1nxi,σ^2=1n∑i=1n(xi−)2.{displaystyle {hat {mu }}={overline {x}}equiv {frac {1}{n}}sum _{i=1}^{n}x_{i},qquad {hat {sigma }}^{2}={frac {1}{n}}sum _{i=1}^{n}(x_{i}-{overline {x}})^{2}.}{hat {mu }}={overline {x}}equiv {frac {1}{n}}sum _{i=1}^{n}x_{i},qquad {hat {sigma }}^{2}={frac {1}{n}}sum _{i=1}^{n}(x_{i}-{overline {x}})^{2}.


Sample mean



Estimator μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} is called the sample mean, since it is the arithmetic mean of all observations. The statistic {displaystyle textstyle {overline {x}}}{displaystyle textstyle {overline {x}}} is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} is the uniformly minimum variance unbiased (UMVU) estimator.[47] In finite samples it is distributed normally:


μ^N(μ2/n).{displaystyle {hat {mu }}sim {mathcal {N}}(mu ,sigma ^{2}/n).}{displaystyle {hat {mu }}sim {mathcal {N}}(mu ,sigma ^{2}/n).}

The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix I−1{displaystyle textstyle {mathcal {I}}^{-1}}{displaystyle textstyle {mathcal {I}}^{-1}}. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} is proportional to 1/n{displaystyle textstyle 1/{sqrt {n}}}{displaystyle textstyle 1/{sqrt {n}}}, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.


From the standpoint of the asymptotic theory, μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} is consistent, that is, it converges in probability to μ as n → ∞. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:


n(μ^μ)→dN(0,σ2).{displaystyle {sqrt {n}}({hat {mu }}-mu ),{xrightarrow {d}},{mathcal {N}}(0,sigma ^{2}).}{displaystyle {sqrt {n}}({hat {mu }}-mu ),{xrightarrow {d}},{mathcal {N}}(0,sigma ^{2}).}


Sample variance



The estimator σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} is called the sample variance, since it is the variance of the sample (x1, …, xn). In practice, another estimator is often used instead of the σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}}. This other estimator is denoted s2, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. The estimator s2 differs from σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} by having (n − 1) instead of n in the denominator (the so-called Bessel's correction):


s2=nn−^2=1n−1∑i=1n(xi−)2.{displaystyle s^{2}={frac {n}{n-1}}{hat {sigma }}^{2}={frac {1}{n-1}}sum _{i=1}^{n}(x_{i}-{overline {x}})^{2}.}{displaystyle s^{2}={frac {n}{n-1}}{hat {sigma }}^{2}={frac {1}{n-1}}sum _{i=1}^{n}(x_{i}-{overline {x}})^{2}.}

The difference between s2 and σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} becomes negligibly small for large n's. In finite samples however, the motivation behind the use of s2 is that it is an unbiased estimator of the underlying parameter σ2, whereas σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} is biased. Also, by the Lehmann–Scheffé theorem the estimator s2 is uniformly minimum variance unbiased (UMVU),[47] which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} is "better" than the s2 in terms of the mean squared error (MSE) criterion. In finite samples both s2 and σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} have scaled chi-squared distribution with (n − 1) degrees of freedom:


s2∼σ2n−1⋅χn−12,σ^2∼σ2n⋅χn−12.{displaystyle s^{2}sim {frac {sigma ^{2}}{n-1}}cdot chi _{n-1}^{2},qquad {hat {sigma }}^{2}sim {frac {sigma ^{2}}{n}}cdot chi _{n-1}^{2}.}{displaystyle s^{2}sim {frac {sigma ^{2}}{n-1}}cdot chi _{n-1}^{2},qquad {hat {sigma }}^{2}sim {frac {sigma ^{2}}{n}}cdot chi _{n-1}^{2}.}

The first of these expressions shows that the variance of s2 is equal to 2σ4/(n−1), which is slightly greater than the σσ-element of the inverse Fisher information matrix I−1{displaystyle textstyle {mathcal {I}}^{-1}}{displaystyle textstyle {mathcal {I}}^{-1}}. Thus, s2 is not an efficient estimator for σ2, and moreover, since s2 is UMVU, we can conclude that the finite-sample efficient estimator for σ2 does not exist.


Applying the asymptotic theory, both estimators s2 and σ^2{displaystyle textstyle {hat {sigma }}^{2}}{displaystyle textstyle {hat {sigma }}^{2}} are consistent, that is they converge in probability to σ2 as the sample size n → ∞. The two estimators are also both asymptotically normal:


n(σ^2−σ2)≃n(s2−σ2)→dN(0,2σ4).{displaystyle {sqrt {n}}({hat {sigma }}^{2}-sigma ^{2})simeq {sqrt {n}}(s^{2}-sigma ^{2}),{xrightarrow {d}},{mathcal {N}}(0,2sigma ^{4}).}{displaystyle {sqrt {n}}({hat {sigma }}^{2}-sigma ^{2})simeq {sqrt {n}}(s^{2}-sigma ^{2}),{xrightarrow {d}},{mathcal {N}}(0,2sigma ^{4}).}

In particular, both estimators are asymptotically efficient for σ2.



Confidence intervals



By Cochran's theorem, for normal distributions the sample mean μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} and the sample variance s2 are independent, which means there can be no gain in considering their joint distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} and s can be employed to construct the so-called t-statistic:


t=μ^μs/n=x¯μ1n(n−1)∑(xi−)2∼tn−1{displaystyle t={frac {{hat {mu }}-mu }{s/{sqrt {n}}}}={frac {{overline {x}}-mu }{sqrt {{frac {1}{n(n-1)}}sum (x_{i}-{overline {x}})^{2}}}}sim t_{n-1}}{displaystyle t={frac {{hat {mu }}-mu }{s/{sqrt {n}}}}={frac {{overline {x}}-mu }{sqrt {{frac {1}{n(n-1)}}sum (x_{i}-{overline {x}})^{2}}}}sim t_{n-1}}

This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;[48] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[49]



μ^tn−1,1−α/21ns,μ^+tn−1,1−α/21ns]≈^|zα/2|1ns,μ^+|zα/2|1ns],{displaystyle mu in left[{hat {mu }}-t_{n-1,1-alpha /2}{frac {1}{sqrt {n}}}s,{hat {mu }}+t_{n-1,1-alpha /2}{frac {1}{sqrt {n}}}sright]approx left[{hat {mu }}-|z_{alpha /2}|{frac {1}{sqrt {n}}}s,{hat {mu }}+|z_{alpha /2}|{frac {1}{sqrt {n}}}sright],}{displaystyle mu in left[{hat {mu }}-t_{n-1,1-alpha /2}{frac {1}{sqrt {n}}}s,{hat {mu }}+t_{n-1,1-alpha /2}{frac {1}{sqrt {n}}}sright]approx left[{hat {mu }}-|z_{alpha /2}|{frac {1}{sqrt {n}}}s,{hat {mu }}+|z_{alpha /2}|{frac {1}{sqrt {n}}}sright],}

σ2∈[(n−1)s2χn−1,1−α/22,(n−1)s2χn−1,α/22]≈[s2−|zα/2|2ns2,s2+|zα/2|2ns2],{displaystyle sigma ^{2}in left[{frac {(n-1)s^{2}}{chi _{n-1,1-alpha /2}^{2}}},{frac {(n-1)s^{2}}{chi _{n-1,alpha /2}^{2}}}right]approx left[s^{2}-|z_{alpha /2}|{frac {sqrt {2}}{sqrt {n}}}s^{2},s^{2}+|z_{alpha /2}|{frac {sqrt {2}}{sqrt {n}}}s^{2}right],}{displaystyle sigma ^{2}in left[{frac {(n-1)s^{2}}{chi _{n-1,1-alpha /2}^{2}}},{frac {(n-1)s^{2}}{chi _{n-1,alpha /2}^{2}}}right]approx left[s^{2}-|z_{alpha /2}|{frac {sqrt {2}}{sqrt {n}}}s^{2},s^{2}+|z_{alpha /2}|{frac {sqrt {2}}{sqrt {n}}}s^{2}right],}


where tk,p and χ 2
k,p
 
are the pth quantiles of the t- and χ2-distributions respectively. These confidence intervals are of the confidence level 1 − α, meaning that the true values μ and σ2 fall outside of these intervals with probability (or significance level) α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The approximate formulas in the display above were derived from the asymptotic distributions of μ^{displaystyle textstyle {hat {mu }}}{displaystyle textstyle {hat {mu }}} and s2. The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles zα/2 do not depend on n. In particular, the most popular value of α = 5%, results in |z0.025| = 1.96.



Bayesian analysis of the normal distribution


Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:



  • Either the mean, or the variance, or neither, may be considered a fixed quantity.

  • When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.

  • Both univariate and multivariate cases need to be considered.

  • Either conjugate or improper prior distributions may be placed on the unknown variables.

  • An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data, but more complex.


The formulas for the non-linear-regression cases are summarized in the conjugate prior article.



Sum of two quadratics



Scalar form


The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.


a(x−y)2+b(x−z)2=(a+b)(x−ay+bza+b)2+aba+b(y−z)2{displaystyle a(x-y)^{2}+b(x-z)^{2}=(a+b)left(x-{frac {ay+bz}{a+b}}right)^{2}+{frac {ab}{a+b}}(y-z)^{2}}a(x-y)^{2}+b(x-z)^{2}=(a+b)left(x-{frac {ay+bz}{a+b}}right)^{2}+{frac {ab}{a+b}}(y-z)^{2}

This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:



  1. The factor ay+bza+b{displaystyle {frac {ay+bz}{a+b}}}{frac {ay+bz}{a+b}} has the form of a weighted average of y and z.


  2. aba+b=11a+1b=(a−1+b−1)−1.{displaystyle {frac {ab}{a+b}}={frac {1}{{frac {1}{a}}+{frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.}{frac {ab}{a+b}}={frac {1}{{frac {1}{a}}+{frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}. This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that aba+b{displaystyle {frac {ab}{a+b}}}{frac {ab}{a+b}} is one-half the harmonic mean of a and b.



Vector form


A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size k{displaystyle ktimes k}ktimes k, then


(y−x)′A(y−x)+(x−z)′B(x−z)=(x−c)′(A+B)(x−c)+(y−z)′(A−1+B−1)−1(y−z){displaystyle {begin{aligned}&(mathbf {y} -mathbf {x} )'mathbf {A} (mathbf {y} -mathbf {x} )+(mathbf {x} -mathbf {z} )'mathbf {B} (mathbf {x} -mathbf {z} )\={}&(mathbf {x} -mathbf {c} )'(mathbf {A} +mathbf {B} )(mathbf {x} -mathbf {c} )+(mathbf {y} -mathbf {z} )'(mathbf {A} ^{-1}+mathbf {B} ^{-1})^{-1}(mathbf {y} -mathbf {z} )end{aligned}}}{displaystyle {begin{aligned}&(mathbf {y} -mathbf {x} )'mathbf {A} (mathbf {y} -mathbf {x} )+(mathbf {x} -mathbf {z} )'mathbf {B} (mathbf {x} -mathbf {z} )\={}&(mathbf {x} -mathbf {c} )'(mathbf {A} +mathbf {B} )(mathbf {x} -mathbf {c} )+(mathbf {y} -mathbf {z} )'(mathbf {A} ^{-1}+mathbf {B} ^{-1})^{-1}(mathbf {y} -mathbf {z} )end{aligned}}}

where


c=(A+B)−1(Ay+Bz){displaystyle mathbf {c} =(mathbf {A} +mathbf {B} )^{-1}(mathbf {A} mathbf {y} +mathbf {B} mathbf {z} )}{displaystyle mathbf {c} =(mathbf {A} +mathbf {B} )^{-1}(mathbf {A} mathbf {y} +mathbf {B} mathbf {z} )}

Note that the form xA x is called a quadratic form and is a scalar:


x′Ax=∑i,jaijxixj{displaystyle mathbf {x} 'mathbf {A} mathbf {x} =sum _{i,j}a_{ij}x_{i}x_{j}}mathbf {x} 'mathbf {A} mathbf {x} =sum _{i,j}a_{ij}x_{i}x_{j}

In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since xixj=xjxi{displaystyle x_{i}x_{j}=x_{j}x_{i}}x_{i}x_{j}=x_{j}x_{i}, only the sum aij+aji{displaystyle a_{ij}+a_{ji}}a_{ij}+a_{ji} matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form x′Ay=y′Ax.{displaystyle mathbf {x} 'mathbf {A} mathbf {y} =mathbf {y} 'mathbf {A} mathbf {x} .}{displaystyle mathbf {x} 'mathbf {A} mathbf {y} =mathbf {y} 'mathbf {A} mathbf {x} .}



Sum of differences from the mean


Another useful formula is as follows:


i=1n(xi−μ)2=∑i=1n(xi−)2+n(x¯μ)2{displaystyle sum _{i=1}^{n}(x_{i}-mu )^{2}=sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}}sum _{i=1}^{n}(x_{i}-mu )^{2}=sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}

where =1n∑i=1nxi.{displaystyle {bar {x}}={frac {1}{n}}sum _{i=1}^{n}x_{i}.}{bar {x}}={frac {1}{n}}sum _{i=1}^{n}x_{i}.



With known variance


For a set of i.i.d. normally distributed data points X of size n where each individual point x follows x∼N(μ2){displaystyle xsim {mathcal {N}}(mu ,sigma ^{2})}xsim {mathcal {N}}(mu ,sigma ^{2}) with known variance σ2, the conjugate prior distribution is also normally distributed.


This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ2. Then if x∼N(μ,1/τ){displaystyle xsim {mathcal {N}}(mu ,1/tau )}xsim {mathcal {N}}(mu ,1/tau ) and μN(μ0,1/τ0),{displaystyle mu sim {mathcal {N}}(mu _{0},1/tau _{0}),}mu sim {mathcal {N}}(mu _{0},1/tau _{0}), we proceed as follows.


First, the likelihood function is (using the formula above for the sum of differences from the mean):


p(X∣μ)=∏i=1nτexp⁡(−12τ(xi−μ)2)=(τ)n/2exp⁡(−12τi=1n(xi−μ)2)=(τ)n/2exp⁡[−12τ(∑i=1n(xi−)2+n(x¯μ)2)].{displaystyle {begin{aligned}p(mathbf {X} mid mu ,tau )&=prod _{i=1}^{n}{sqrt {frac {tau }{2pi }}}exp left(-{frac {1}{2}}tau (x_{i}-mu )^{2}right)\&=left({frac {tau }{2pi }}right)^{n/2}exp left(-{frac {1}{2}}tau sum _{i=1}^{n}(x_{i}-mu )^{2}right)\&=left({frac {tau }{2pi }}right)^{n/2}exp left[-{frac {1}{2}}tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right].end{aligned}}}{displaystyle {begin{aligned}p(mathbf {X} mid mu ,tau )&=prod _{i=1}^{n}{sqrt {frac {tau }{2pi }}}exp left(-{frac {1}{2}}tau (x_{i}-mu )^{2}right)\&=left({frac {tau }{2pi }}right)^{n/2}exp left(-{frac {1}{2}}tau sum _{i=1}^{n}(x_{i}-mu )^{2}right)\&=left({frac {tau }{2pi }}right)^{n/2}exp left[-{frac {1}{2}}tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right].end{aligned}}}

Then, we proceed as follows:


p(μX)∝p(X∣μ)p(μ)=(τ)n/2exp⁡[−12τ(∑i=1n(xi−)2+n(x¯μ)2)]τ02πexp⁡(−12τ0(μμ0)2)∝exp⁡(−12(τ(∑i=1n(xi−)2+n(x¯μ)2)+τ0(μμ0)2))∝exp⁡(−12(nτ(x¯μ)2+τ0(μμ0)2))=exp⁡(−12(nτ0)(μ0nτ0)2+nττ0nτ0(x¯μ0)2)∝exp⁡(−12(nτ0)(μ0nτ0)2){displaystyle {begin{aligned}p(mu mid mathbf {X} )&propto p(mathbf {X} mid mu )p(mu )\&=left({frac {tau }{2pi }}right)^{n/2}exp left[-{frac {1}{2}}tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right]{sqrt {frac {tau _{0}}{2pi }}}exp left(-{frac {1}{2}}tau _{0}(mu -mu _{0})^{2}right)\&propto exp left(-{frac {1}{2}}left(tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)+tau _{0}(mu -mu _{0})^{2}right)right)\&propto exp left(-{frac {1}{2}}left(ntau ({bar {x}}-mu )^{2}+tau _{0}(mu -mu _{0})^{2}right)right)\&=exp left(-{frac {1}{2}}(ntau +tau _{0})left(mu -{dfrac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}right)^{2}+{frac {ntau tau _{0}}{ntau +tau _{0}}}({bar {x}}-mu _{0})^{2}right)\&propto exp left(-{frac {1}{2}}(ntau +tau _{0})left(mu -{dfrac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}right)^{2}right)end{aligned}}}{displaystyle {begin{aligned}p(mu mid mathbf {X} )&propto p(mathbf {X} mid mu )p(mu )\&=left({frac {tau }{2pi }}right)^{n/2}exp left[-{frac {1}{2}}tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right]{sqrt {frac {tau _{0}}{2pi }}}exp left(-{frac {1}{2}}tau _{0}(mu -mu _{0})^{2}right)\&propto exp left(-{frac {1}{2}}left(tau left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)+tau _{0}(mu -mu _{0})^{2}right)right)\&propto exp left(-{frac {1}{2}}left(ntau ({bar {x}}-mu )^{2}+tau _{0}(mu -mu _{0})^{2}right)right)\&=exp left(-{frac {1}{2}}(ntau +tau _{0})left(mu -{dfrac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}right)^{2}+{frac {ntau tau _{0}}{ntau +tau _{0}}}({bar {x}}-mu _{0})^{2}right)\&propto exp left(-{frac {1}{2}}(ntau +tau _{0})left(mu -{dfrac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}right)^{2}right)end{aligned}}}

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean 0nτ0{displaystyle {frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}}{frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}} and precision 0{displaystyle ntau +tau _{0}}ntau +tau _{0}, i.e.


p(μX)∼N(nτ0nτ0,1nτ0){displaystyle p(mu mid mathbf {X} )sim {mathcal {N}}left({frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}},{frac {1}{ntau +tau _{0}}}right)}p(mu mid mathbf {X} )sim {mathcal {N}}left({frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}},{frac {1}{ntau +tau _{0}}}right)

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:


τ0′=τ0+nτμ0′=nτ0nτ0x¯=1n∑i=1nxi{displaystyle {begin{aligned}tau _{0}'&=tau _{0}+ntau \mu _{0}'&={frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}\{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}end{aligned}}}{begin{aligned}tau _{0}'&=tau _{0}+ntau \mu _{0}'&={frac {ntau {bar {x}}+tau _{0}mu _{0}}{ntau +tau _{0}}}\{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}end{aligned}}

That is, to combine n data points with total precision of (or equivalently, total variance of n/σ2) and mean of values {displaystyle {bar {x}}}{bar {x}}, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)


The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas


σ02′=1nσ2+1σ02μ0′=nx¯σ2+μ02nσ2+1σ02x¯=1n∑i=1nxi{displaystyle {begin{aligned}{sigma _{0}^{2}}'&={frac {1}{{frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}}}\mu _{0}'&={frac {{frac {n{bar {x}}}{sigma ^{2}}}+{frac {mu _{0}}{sigma _{0}^{2}}}}{{frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}}}\{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}end{aligned}}}{begin{aligned}{sigma _{0}^{2}}'&={frac {1}{{frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}}}\mu _{0}'&={frac {{frac {n{bar {x}}}{sigma ^{2}}}+{frac {mu _{0}}{sigma _{0}^{2}}}}{{frac {n}{sigma ^{2}}}+{frac {1}{sigma _{0}^{2}}}}}\{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}end{aligned}}


With known mean


For a set of i.i.d. normally distributed data points X of size n where each individual point x follows x∼N(μ2){displaystyle xsim {mathcal {N}}(mu ,sigma ^{2})}xsim {mathcal {N}}(mu ,sigma ^{2}) with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ2 is as follows:


p(σ2∣ν0,σ02)=(σ02ν02)ν0/2Γ02) exp⁡[−ν022σ2](σ2)1+ν02∝exp⁡[−ν022σ2](σ2)1+ν02{displaystyle p(sigma ^{2}mid nu _{0},sigma _{0}^{2})={frac {(sigma _{0}^{2}{frac {nu _{0}}{2}})^{nu _{0}/2}}{Gamma left({frac {nu _{0}}{2}}right)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}propto {frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}}{displaystyle p(sigma ^{2}mid nu _{0},sigma _{0}^{2})={frac {(sigma _{0}^{2}{frac {nu _{0}}{2}})^{nu _{0}/2}}{Gamma left({frac {nu _{0}}{2}}right)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}propto {frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}}

The likelihood function from above, written in terms of the variance, is:


p(X∣μ2)=(12πσ2)n/2exp⁡[−12σ2∑i=1n(xi−μ)2]=(12πσ2)n/2exp⁡[−S2σ2]{displaystyle {begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}sum _{i=1}^{n}(x_{i}-mu )^{2}right]\&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {S}{2sigma ^{2}}}right]end{aligned}}}{displaystyle {begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}sum _{i=1}^{n}(x_{i}-mu )^{2}right]\&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {S}{2sigma ^{2}}}right]end{aligned}}}

where


S=∑i=1n(xi−μ)2.{displaystyle S=sum _{i=1}^{n}(x_{i}-mu )^{2}.}S=sum _{i=1}^{n}(x_{i}-mu )^{2}.

Then:


p(σ2∣X)∝p(X∣σ2)p(σ2)=(12πσ2)n/2exp⁡[−S2σ2](σ02ν02)ν02Γ02) exp⁡[−ν022σ2](σ2)1+ν02∝(1σ2)n/21(σ2)1+ν02exp⁡[−S2σ2+−ν022σ2]=1(σ2)1+ν0+n2exp⁡[−ν02+S2σ2]{displaystyle {begin{aligned}p(sigma ^{2}mid mathbf {X} )&propto p(mathbf {X} mid sigma ^{2})p(sigma ^{2})\&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {S}{2sigma ^{2}}}right]{frac {(sigma _{0}^{2}{frac {nu _{0}}{2}})^{frac {nu _{0}}{2}}}{Gamma left({frac {nu _{0}}{2}}right)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}\&propto left({frac {1}{sigma ^{2}}}right)^{n/2}{frac {1}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}exp left[-{frac {S}{2sigma ^{2}}}+{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]\&={frac {1}{(sigma ^{2})^{1+{frac {nu _{0}+n}{2}}}}}exp left[-{frac {nu _{0}sigma _{0}^{2}+S}{2sigma ^{2}}}right]end{aligned}}}{displaystyle {begin{aligned}p(sigma ^{2}mid mathbf {X} )&propto p(mathbf {X} mid sigma ^{2})p(sigma ^{2})\&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {S}{2sigma ^{2}}}right]{frac {(sigma _{0}^{2}{frac {nu _{0}}{2}})^{frac {nu _{0}}{2}}}{Gamma left({frac {nu _{0}}{2}}right)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}\&propto left({frac {1}{sigma ^{2}}}right)^{n/2}{frac {1}{(sigma ^{2})^{1+{frac {nu _{0}}{2}}}}}exp left[-{frac {S}{2sigma ^{2}}}+{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]\&={frac {1}{(sigma ^{2})^{1+{frac {nu _{0}+n}{2}}}}}exp left[-{frac {nu _{0}sigma _{0}^{2}+S}{2sigma ^{2}}}right]end{aligned}}}

The above is also a scaled inverse chi-squared distribution where


ν0′=ν0+nν0′σ02′=ν02+∑i=1n(xi−μ)2{displaystyle {begin{aligned}nu _{0}'&=nu _{0}+n\nu _{0}'{sigma _{0}^{2}}'&=nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-mu )^{2}end{aligned}}}{begin{aligned}nu _{0}'&=nu _{0}+n\nu _{0}'{sigma _{0}^{2}}'&=nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-mu )^{2}end{aligned}}

or equivalently


ν0′=ν0+nσ02′=ν02+∑i=1n(xi−μ)2ν0+n{displaystyle {begin{aligned}nu _{0}'&=nu _{0}+n\{sigma _{0}^{2}}'&={frac {nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-mu )^{2}}{nu _{0}+n}}end{aligned}}}{begin{aligned}nu _{0}'&=nu _{0}+n\{sigma _{0}^{2}}'&={frac {nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-mu )^{2}}{nu _{0}+n}}end{aligned}}

Reparameterizing in terms of an inverse gamma distribution, the result is:


α′=α+n2β′=β+∑i=1n(xi−μ)22{displaystyle {begin{aligned}alpha '&=alpha +{frac {n}{2}}\beta '&=beta +{frac {sum _{i=1}^{n}(x_{i}-mu )^{2}}{2}}end{aligned}}}{begin{aligned}alpha '&=alpha +{frac {n}{2}}\beta '&=beta +{frac {sum _{i=1}^{n}(x_{i}-mu )^{2}}{2}}end{aligned}}


With unknown mean and unknown variance


For a set of i.i.d. normally distributed data points X of size n where each individual point x follows x∼N(μ2){displaystyle xsim {mathcal {N}}(mu ,sigma ^{2})}xsim {mathcal {N}}(mu ,sigma ^{2}) with unknown mean μ and unknown variance σ2, a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:



  1. From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.

  2. From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.

  3. Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.

  4. To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.

  5. This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Note that each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.

  6. This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.


The priors are normally defined as follows:


p(μσ2;μ0,n0)∼N(μ0,σ2/n0)p(σ2;ν0,σ02)∼2(ν0,σ02)=IG(ν0/2,ν02/2){displaystyle {begin{aligned}p(mu mid sigma ^{2};mu _{0},n_{0})&sim {mathcal {N}}(mu _{0},sigma ^{2}/n_{0})\p(sigma ^{2};nu _{0},sigma _{0}^{2})&sim Ichi ^{2}(nu _{0},sigma _{0}^{2})=IG(nu _{0}/2,nu _{0}sigma _{0}^{2}/2)end{aligned}}}{begin{aligned}p(mu mid sigma ^{2};mu _{0},n_{0})&sim {mathcal {N}}(mu _{0},sigma ^{2}/n_{0})\p(sigma ^{2};nu _{0},sigma _{0}^{2})&sim Ichi ^{2}(nu _{0},sigma _{0}^{2})=IG(nu _{0}/2,nu _{0}sigma _{0}^{2}/2)end{aligned}}

The update equations can be derived, and look as follows:


=1n∑i=1nxiμ0′=n0μ0+nx¯n0+nn0′=n0+nν0′=ν0+nν0′σ02′=ν02+∑i=1n(xi−)2+n0nn0+n(μ0−)2{displaystyle {begin{aligned}{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}\mu _{0}'&={frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}\n_{0}'&=n_{0}+n\nu _{0}'&=nu _{0}+n\nu _{0}'{sigma _{0}^{2}}'&=nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}end{aligned}}}{displaystyle {begin{aligned}{bar {x}}&={frac {1}{n}}sum _{i=1}^{n}x_{i}\mu _{0}'&={frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}\n_{0}'&=n_{0}+n\nu _{0}'&=nu _{0}+n\nu _{0}'{sigma _{0}^{2}}'&=nu _{0}sigma _{0}^{2}+sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}end{aligned}}}

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for ν0′σ02′{displaystyle nu _{0}'{sigma _{0}^{2}}'}nu _{0}'{sigma _{0}^{2}}' is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new "interaction term" needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.



[Proof]


The prior distributions are


p(μσ2;μ0,n0)∼N(μ0,σ2/n0)=12πσ2n0exp⁡(−n02σ2(μμ0)2)∝2)−1/2exp⁡(−n02σ2(μμ0)2)p(σ2;ν0,σ02)∼2(ν0,σ02)=IG(ν0/2,ν02/2)=(σ02ν0/2)ν0/2Γ0/2) exp⁡[−ν022σ2](σ2)1+ν0/2∝2)−(1+ν0/2)exp⁡[−ν022σ2].{displaystyle {begin{aligned}p(mu mid sigma ^{2};mu _{0},n_{0})&sim {mathcal {N}}(mu _{0},sigma ^{2}/n_{0})={frac {1}{sqrt {2pi {frac {sigma ^{2}}{n_{0}}}}}}exp left(-{frac {n_{0}}{2sigma ^{2}}}(mu -mu _{0})^{2}right)\&propto (sigma ^{2})^{-1/2}exp left(-{frac {n_{0}}{2sigma ^{2}}}(mu -mu _{0})^{2}right)\p(sigma ^{2};nu _{0},sigma _{0}^{2})&sim Ichi ^{2}(nu _{0},sigma _{0}^{2})=IG(nu _{0}/2,nu _{0}sigma _{0}^{2}/2)\&={frac {(sigma _{0}^{2}nu _{0}/2)^{nu _{0}/2}}{Gamma (nu _{0}/2)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+nu _{0}/2}}}\&propto {(sigma ^{2})^{-(1+nu _{0}/2)}}exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right].end{aligned}}}{displaystyle {begin{aligned}p(mu mid sigma ^{2};mu _{0},n_{0})&sim {mathcal {N}}(mu _{0},sigma ^{2}/n_{0})={frac {1}{sqrt {2pi {frac {sigma ^{2}}{n_{0}}}}}}exp left(-{frac {n_{0}}{2sigma ^{2}}}(mu -mu _{0})^{2}right)\&propto (sigma ^{2})^{-1/2}exp left(-{frac {n_{0}}{2sigma ^{2}}}(mu -mu _{0})^{2}right)\p(sigma ^{2};nu _{0},sigma _{0}^{2})&sim Ichi ^{2}(nu _{0},sigma _{0}^{2})=IG(nu _{0}/2,nu _{0}sigma _{0}^{2}/2)\&={frac {(sigma _{0}^{2}nu _{0}/2)^{nu _{0}/2}}{Gamma (nu _{0}/2)}}~{frac {exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right]}{(sigma ^{2})^{1+nu _{0}/2}}}\&propto {(sigma ^{2})^{-(1+nu _{0}/2)}}exp left[{frac {-nu _{0}sigma _{0}^{2}}{2sigma ^{2}}}right].end{aligned}}}

Therefore, the joint prior is


p(μ2;μ0,n0,ν0,σ02)=p(μσ2;μ0,n0)p(σ2;ν0,σ02)∝2)−0+3)/2exp⁡[−12σ2(ν02+n0(μμ0)2)].{displaystyle {begin{aligned}p(mu ,sigma ^{2};mu _{0},n_{0},nu _{0},sigma _{0}^{2})&=p(mu mid sigma ^{2};mu _{0},n_{0}),p(sigma ^{2};nu _{0},sigma _{0}^{2})\&propto (sigma ^{2})^{-(nu _{0}+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+n_{0}(mu -mu _{0})^{2}right)right].end{aligned}}}{displaystyle {begin{aligned}p(mu ,sigma ^{2};mu _{0},n_{0},nu _{0},sigma _{0}^{2})&=p(mu mid sigma ^{2};mu _{0},n_{0}),p(sigma ^{2};nu _{0},sigma _{0}^{2})\&propto (sigma ^{2})^{-(nu _{0}+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+n_{0}(mu -mu _{0})^{2}right)right].end{aligned}}}

The likelihood function from the section above with known variance is:


p(X∣μ2)=(12πσ2)n/2exp⁡[−12σ2(∑i=1n(xi−μ)2)]{displaystyle {begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}left(sum _{i=1}^{n}(x_{i}-mu )^{2}right)right]end{aligned}}}{begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}left(sum _{i=1}^{n}(x_{i}-mu )^{2}right)right]end{aligned}}

Writing it in terms of variance rather than precision, we get:


p(X∣μ2)=(12πσ2)n/2exp⁡[−12σ2(∑i=1n(xi−)2+n(x¯μ)2)]∝σ2−n/2exp⁡[−12σ2(S+n(x¯μ)2)]{displaystyle {begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right]\&propto {sigma ^{2}}^{-n/2}exp left[-{frac {1}{2sigma ^{2}}}left(S+n({bar {x}}-mu )^{2}right)right]end{aligned}}}{begin{aligned}p(mathbf {X} mid mu ,sigma ^{2})&=left({frac {1}{2pi sigma ^{2}}}right)^{n/2}exp left[-{frac {1}{2sigma ^{2}}}left(sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}+n({bar {x}}-mu )^{2}right)right]\&propto {sigma ^{2}}^{-n/2}exp left[-{frac {1}{2sigma ^{2}}}left(S+n({bar {x}}-mu )^{2}right)right]end{aligned}}

where S=∑i=1n(xi−)2.{displaystyle S=sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}.}S=sum _{i=1}^{n}(x_{i}-{bar {x}})^{2}.


Therefore, the posterior is (dropping the hyperparameters as conditioning factors):


p(μ2∣X)∝p(μ2)p(X∣μ2)∝2)−0+3)/2exp⁡[−12σ2(ν02+n0(μμ0)2)]σ2−n/2exp⁡[−12σ2(S+n(x¯μ)2)]=(σ2)−0+n+3)/2exp⁡[−12σ2(ν02+S+n0(μμ0)2+n(x¯μ)2)]=(σ2)−0+n+3)/2exp⁡[−12σ2(ν02+S+n0nn0+n(μ0−)2+(n0+n)(μn0μ0+nx¯n0+n)2)]∝2)−1/2exp⁡[−n0+n2σ2(μn0μ0+nx¯n0+n)2]×2)−0/2+n/2+1)exp⁡[−12σ2(ν02+S+n0nn0+n(μ0−)2)]=Nμσ2(n0μ0+nx¯n0+n,σ2n0+n)⋅IGσ2(12(ν0+n),12(ν02+S+n0nn0+n(μ0−)2)).{displaystyle {begin{aligned}p(mu ,sigma ^{2}mid mathbf {X} )&propto p(mu ,sigma ^{2}),p(mathbf {X} mid mu ,sigma ^{2})\&propto (sigma ^{2})^{-(nu _{0}+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+n_{0}(mu -mu _{0})^{2}right)right]{sigma ^{2}}^{-n/2}exp left[-{frac {1}{2sigma ^{2}}}left(S+n({bar {x}}-mu )^{2}right)right]\&=(sigma ^{2})^{-(nu _{0}+n+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+n_{0}(mu -mu _{0})^{2}+n({bar {x}}-mu )^{2}right)right]\&=(sigma ^{2})^{-(nu _{0}+n+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}+(n_{0}+n)left(mu -{frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}right)^{2}right)right]\&propto (sigma ^{2})^{-1/2}exp left[-{frac {n_{0}+n}{2sigma ^{2}}}left(mu -{frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}right)^{2}right]\&quad times (sigma ^{2})^{-(nu _{0}/2+n/2+1)}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}right)right]\&={mathcal {N}}_{mu mid sigma ^{2}}left({frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}},{frac {sigma ^{2}}{n_{0}+n}}right)cdot {rm {IG}}_{sigma ^{2}}left({frac {1}{2}}(nu _{0}+n),{frac {1}{2}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}right)right).end{aligned}}}{begin{aligned}p(mu ,sigma ^{2}mid mathbf {X} )&propto p(mu ,sigma ^{2}),p(mathbf {X} mid mu ,sigma ^{2})\&propto (sigma ^{2})^{-(nu _{0}+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+n_{0}(mu -mu _{0})^{2}right)right]{sigma ^{2}}^{-n/2}exp left[-{frac {1}{2sigma ^{2}}}left(S+n({bar {x}}-mu )^{2}right)right]\&=(sigma ^{2})^{-(nu _{0}+n+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+n_{0}(mu -mu _{0})^{2}+n({bar {x}}-mu )^{2}right)right]\&=(sigma ^{2})^{-(nu _{0}+n+3)/2}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}+(n_{0}+n)left(mu -{frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}right)^{2}right)right]\&propto (sigma ^{2})^{-1/2}exp left[-{frac {n_{0}+n}{2sigma ^{2}}}left(mu -{frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}}right)^{2}right]\&quad times (sigma ^{2})^{-(nu _{0}/2+n/2+1)}exp left[-{frac {1}{2sigma ^{2}}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}right)right]\&={mathcal {N}}_{mu mid sigma ^{2}}left({frac {n_{0}mu _{0}+n{bar {x}}}{n_{0}+n}},{frac {sigma ^{2}}{n_{0}+n}}right)cdot {rm {IG}}_{sigma ^{2}}left({frac {1}{2}}(nu _{0}+n),{frac {1}{2}}left(nu _{0}sigma _{0}^{2}+S+{frac {n_{0}n}{n_{0}+n}}(mu _{0}-{bar {x}})^{2}right)right).end{aligned}}

In other words, the posterior distribution has the form of a product of a normal distribution over p(μ | σ2) times an inverse gamma distribution over p2), with parameters that are the same as the update equations above.





Occurrence and applications


The occurrence of normal distribution in practical problems can be loosely classified into four categories:



  1. Exactly normal distributions;

  2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and

  3. Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.

  4. Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.



Exact normality




The ground state of a quantum harmonic oscillator has the Gaussian distribution.


Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:



  • Probability density function of a ground state in a quantum harmonic oscillator.

  • The position of a particle that experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is the dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation Template:Sfrac2 f(x,t) = Template:Sfrac2 Template:Sfrac2 f(x,t). If the initial location is given by a certain density function g(x), then the density at time t is the convolution of g and the normal PDF.



Approximate normality


Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.



  • In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as


    • Binomial random variables, associated with binary response variables;


    • Poisson random variables, associated with rare events;




  • Thermal radiation has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.



Assumed normality




Histogram of sepal widths for Iris versicolor from Fisher's Iris flower data set, with superimposed best-fitting normal distribution.


.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}

I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.


— Pearson (1901)


There are statistical methods to empirically test that assumption, see the above Normality tests section.



  • In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including:

    • Measures of size of living tissue (length, height, skin area, weight);[50]

    • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;

    • Certain physiological measurements, such as blood pressure of adult humans.



  • In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.


  • Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.[51]

  • In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or transforming the raw test scores into "output" scores by fitting them to the normal distribution. For example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.




Fitted cumulative normal distribution to October rainfalls, see distribution fitting



  • Many scores are derived from the normal distribution, including percentile ranks ("percentiles" or "quantiles"), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.

  • In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem.[52] The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.



Produced normality


In regression analysis, lack of normality in residuals simply indicates that the model postulated is inadequate in accounting for the tendency in the data and needs to be augmented; in other words, normality in residuals can always be achieved given a properly constructed model.



Generating values from normal distribution




The bean machine, a device invented by Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.


In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ2
)
can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.



  • The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ−1(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ−1, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places,[53] which is used by R to compute random variates of the normal distribution.

  • An easy to program approximate approach, that relies on the central limit theorem, is as follows: generate 12 uniform U(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).[54]

  • The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1). Then the two random variables X and Y



X=−2ln⁡Ucos⁡(2πV),Y=−2ln⁡Usin⁡(2πV).{displaystyle X={sqrt {-2ln U}},cos(2pi V),qquad Y={sqrt {-2ln U}},sin(2pi V).}X={sqrt {-2ln U}},cos(2pi V),qquad Y={sqrt {-2ln U}},sin(2pi V).

will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector (X, Y) the squared norm X2 + Y2 will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.


  • The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, U and V are drawn from the uniform (−1,1) distribution, and then S = U2 + V2 is computed. If S is greater or equal to 1, then the method starts over, otherwise the two quantities


X=U−2ln⁡SS,Y=V−2ln⁡SS{displaystyle X=U{sqrt {frac {-2ln S}{S}}},qquad Y=V{sqrt {frac {-2ln S}{S}}}}{displaystyle X=U{sqrt {frac {-2ln S}{S}}},qquad Y=V{sqrt {frac {-2ln S}{S}}}}

are returned. Again, X and Y are independent, standard normal random variables.


  • The Ratio method[55] is a rejection method. The algorithm proceeds as follows:

    • Generate two independent uniform deviates U and V;

    • Compute X = 8/e (V − 0.5)/U;

    • Optional: if X2 ≤ 5 − 4e1/4U then accept X and terminate algorithm;

    • Optional: if X2 ≥ 4e−1.35/U + 1.4 then reject X and start over from step 1;

    • If X2 ≤ −4 lnU then accept X, otherwise start over the algorithm.



The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved[56] so that the logarithm is rarely evaluated.


  • The ziggurat algorithm[57] is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.

  • Integer arithmetic can be used to sample from the standard normal distribution.[58] This method is exact in the sense that it satisfies the conditions of ideal approximation;[59] i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.

  • There is also some investigation[60] into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.



Numerical approximations for the normal CDF


The standard normal CDF is widely used in scientific and statistical computing.


The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.




  • Zelen & Severo (1964) give the approximation for Φ(x) for x > 0 with the absolute error |ε(x)| < 7.5·10−8 (algorithm 26.2.17):
    Φ(x)=1−φ(x)(b1t+b2t2+b3t3+b4t4+b5t5)+ε(x),t=11+b0x,{displaystyle Phi (x)=1-varphi (x)left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}right)+varepsilon (x),qquad t={frac {1}{1+b_{0}x}},}{displaystyle Phi (x)=1-varphi (x)left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}right)+varepsilon (x),qquad t={frac {1}{1+b_{0}x}},}

    where ϕ(x) is the standard normal PDF, and b0 = 0.2316419, b1 = 0.319381530, b2 = −0.356563782, b3 = 1.781477937, b4 = −1.821255978, b5 = 1.330274429.


  • Hart (1968) lists some dozens of approximations – by means of rational functions, with or without exponentials – for the .mw-parser-output .monospaced{font-family:monospace,monospace}
    erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.


  • Cody (1969) after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.


  • Marsaglia (2004) suggested a simple algorithm[nb 1] based on the Taylor series expansion

    Φ(x)=12+φ(x)(x+x33+x53⋅5+x73⋅5⋅7+x93⋅5⋅7⋅9+⋯){displaystyle Phi (x)={frac {1}{2}}+varphi (x)left(x+{frac {x^{3}}{3}}+{frac {x^{5}}{3cdot 5}}+{frac {x^{7}}{3cdot 5cdot 7}}+{frac {x^{9}}{3cdot 5cdot 7cdot 9}}+cdots right)}{displaystyle Phi (x)={frac {1}{2}}+varphi (x)left(x+{frac {x^{3}}{3}}+{frac {x^{5}}{3cdot 5}}+{frac {x^{7}}{3cdot 5cdot 7}}+{frac {x^{9}}{3cdot 5cdot 7cdot 9}}+cdots right)}


    for calculating Φ(x) with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when x = 10).

  • The GNU Scientific Library calculates values of the standard normal CDF using Hart's algorithms and approximations with Chebyshev polynomials.


Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting p=Φ(z), the simplest approximation for the quantile function is:


z=Φ1(p)=5.5556[1−(1−pp)0.1186],p≥1/2{displaystyle z=Phi ^{-1}(p)=5.5556left[1-left({frac {1-p}{p}}right)^{0.1186}right],qquad pgeq 1/2}{displaystyle z=Phi ^{-1}(p)=5.5556left[1-left({frac {1-p}{p}}right)^{0.1186}right],qquad pgeq 1/2}

This approximation delivers for z a maximum absolute error of 0.026 (for 0.5 ≤ p ≤ 0.9999, corresponding to 0 ≤ z ≤ 3.719). For p < 1/2 replace p by 1 − p and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:


z=−0.4115{1−pp+log⁡[1−pp]−1},p≥1/2{displaystyle z=-0.4115left{{frac {1-p}{p}}+log left[{frac {1-p}{p}}right]-1right},qquad pgeq 1/2}{displaystyle z=-0.4115left{{frac {1-p}{p}}+log left[{frac {1-p}{p}}right]-1right},qquad pgeq 1/2}

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by


L(z)=∫z∞(u−z)φ(u)du=∫z∞[1−Φ(u)]duL(z)≈{0.4115(p1−p)−z,p<1/2,0.4115(1−pp),p≥1/2.or, equivalently,L(z)≈{0.4115{1−log⁡[p1−p]},p<1/2,0.41151−pp,p≥1/2.{displaystyle {begin{aligned}L(z)&=int _{z}^{infty }(u-z)varphi (u),du=int _{z}^{infty }[1-Phi (u)],du\[5pt]L(z)&approx {begin{cases}0.4115left({dfrac {p}{1-p}}right)-z,&p<1/2,\\0.4115left({dfrac {1-p}{p}}right),&pgeq 1/2.end{cases}}\[5pt]{text{or, equivalently,}}\L(z)&approx {begin{cases}0.4115left{1-log left[{frac {p}{1-p}}right]right},&p<1/2,\\0.4115{dfrac {1-p}{p}},&pgeq 1/2.end{cases}}end{aligned}}}{displaystyle {begin{aligned}L(z)&=int _{z}^{infty }(u-z)varphi (u),du=int _{z}^{infty }[1-Phi (u)],du\[5pt]L(z)&approx {begin{cases}0.4115left({dfrac {p}{1-p}}right)-z,&p<1/2,\\0.4115left({dfrac {1-p}{p}}right),&pgeq 1/2.end{cases}}\[5pt]{text{or, equivalently,}}\L(z)&approx {begin{cases}0.4115left{1-log left[{frac {p}{1-p}}right]right},&p<1/2,\\0.4115{dfrac {1-p}{p}},&pgeq 1/2.end{cases}}end{aligned}}}

This approximation is particularly accurate for the right far-tail (maximum error of 10−3 for z≥1.4). Highly accurate approximations for the CDF, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).


Some more approximations can be found at: Error function#Approximation with elementary functions.



History



Development


Some authors[61][62] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738[nb 2] published in the second edition of his "The Doctrine of Chances" the study of the coefficients in the binomial expansion of (a + b)n. De Moivre proved that the middle term in this expansion has the approximate magnitude of 2/2πn{displaystyle 2/{sqrt {2pi n}}}2/{sqrt {2pi n}}, and that "If m or ½n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval , has to the middle Term, is 2ℓn{displaystyle -{frac {2ell ell }{n}}}-{frac {2ell ell }{n}}."[63] Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.[64]





Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.


In 1809 Gauss published his monograph "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M, M′′, … to denote the measurements of some unknown quantity V, and sought the "most probable" estimator of that quantity: the one that maximizes the probability φ(M − V) · φ(M′ − V) · φ(M′′ − V) · … of obtaining the observed experimental results. In his notation φΔ is the probability law of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.[nb 3] Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:[65]


φΔ=h√πe−hhΔΔ,{displaystyle varphi {mathit {Delta }}={frac {h}{surd pi }},e^{-mathrm {hh} Delta Delta },}{displaystyle varphi {mathit {Delta }}={frac {h}{surd pi }},e^{-mathrm {hh} Delta Delta },}

where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares (NWLS) method.[66]





Marquis de Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.


Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.[nb 4] It was Laplace who first posed the problem of aggregating several observations in 1774,[67] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral et2 dt = π in 1782, providing the normalization constant for the normal distribution.[68] Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.[69]


It is of interest to note that in 1809 an American mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss.[70] His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.[71]


In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:[72] "The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is


N⁡πe−x2α2dx{displaystyle operatorname {N} {frac {1}{alpha ;{sqrt {pi }}}};e^{-{frac {x^{2}}{alpha ^{2}}}},dx}{displaystyle operatorname {N} {frac {1}{alpha ;{sqrt {pi }}}};e^{-{frac {x^{2}}{alpha ^{2}}}},dx}


Naming


Since its introduction, the normal distribution has been known by many different names: the law of error, the law of facility of errors, Laplace's second law, Gaussian law, etc. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual".[73] However, by the end of the 19th century some authors[nb 5] had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."[74] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.[75]




Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'.


— Pearson (1920)


Also, it was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:


df=12σe−(x−m)2/(2σ2)dx{displaystyle df={frac {1}{sqrt {2sigma ^{2}pi }}}e^{-(x-m)^{2}/(2sigma ^{2})},dx}{displaystyle df={frac {1}{sqrt {2sigma ^{2}pi }}}e^{-(x-m)^{2}/(2sigma ^{2})},dx}

The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P.G. Hoel (1947) "Introduction to mathematical statistics" and A.M. Mood (1950) "Introduction to the theory of statistics".[76]


When the name is used, the "Gaussian distribution" was named after Carl Friedrich Gauss, who introduced the distribution in 1809 as a way of rationalizing the method of least squares as outlined above. Among English speakers, both "normal distribution" and "Gaussian distribution" are in common use, with different terms preferred by different communities.



See also





  • Wrapped normal distribution — the Normal distribution applied to a circular domain


  • Bates distribution — similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range


  • Behrens–Fisher problem — the long-standing problem of testing whether two normal samples with different variances have same means;


  • Bhattacharyya distance – method used to separate mixtures of normal distributions


  • Erdős–Kac theorem—on the occurrence of the normal distribution in number theory


  • Gaussian blur—convolution, which uses the normal distribution as a kernel

  • Normally distributed and uncorrelated does not imply independent

  • Sub-Gaussian distribution

  • Sum of normally distributed random variables


  • Tweedie distribution — The normal distribution is a member of the family of Tweedie exponential dispersion models


  • Z-test— using the normal distribution



Notes





  1. ^ For example, this algorithm is given in the article Bc programming language.


  2. ^ De Moivre first published his findings in 1733, in a pamphlet "Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem Expansi" that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example Walker (1985).


  3. ^ "It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — Gauss (1809, section 177)


  4. ^ "My custom of terming the curve the Gauss–Laplacian or normal curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from Pearson (1905, p. 189)


  5. ^ Besides those specifically referenced here, such use is encountered in the works of Peirce, Galton (Galton (1889, chapter V)) and Lexis (Lexis (1878), Rohrbasser & Véron (2003)) c. 1875.[citation needed]




References



Citations





  1. ^ Normal Distribution, Gale Encyclopedia of Psychology


  2. ^ Casella & Berger (2001, p. 102)


  3. ^ Lyon, A. (2014). Why are Normal Distributions Normal?, The British Journal for the Philosophy of Science.


  4. ^ For the proof see Gaussian integral


  5. ^ Stigler (1982)


  6. ^ Halperin, Hartley & Hoel (1965, item 7)


  7. ^ McPherson (1990, p. 110)


  8. ^ Bernardo & Smith (2000, p. 121)


  9. ^ Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory. John Wiley and Sons. p. 254..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


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  11. ^ Geary RC(1936) The distribution of the "Student's" ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184


  12. ^ Lukas E (1942) A characterization of the normal distribution. Annals of Mathematical Statistics 13: 91–93


  13. ^ abc Patel & Read (1996, [2.1.4])


  14. ^ Fan (1991, p. 1258)


  15. ^ Patel & Read (1996, [2.1.8])


  16. ^ Papoulis, Athanasios. Probability, Random Variables and Stochastic Processes (4th Edition). p. 148.


  17. ^ Bryc (1995, p. 23)


  18. ^ Bryc (1995, p. 24)


  19. ^ Scott, Clayton; Nowak, Robert (August 7, 2003). "The Q-function". Connexions.


  20. ^ Barak, Ohad (April 6, 2006). "Q Function and Error Function" (PDF). Tel Aviv University. Archived from the original (PDF) on March 25, 2009.


  21. ^ Weisstein, Eric W. "Normal Distribution Function". MathWorld.


  22. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26, eqn 26.2.12". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 932. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.


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  26. ^ "Normal Approximation to Poisson Distribution". Stat.ucla.edu. Retrieved 2017-03-03.


  27. ^ Cover & Thomas (2006, p. 254)


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  31. ^ Bryc (1995, p. 27)


  32. ^ Patel & Read (1996, [2.3.6])


  33. ^ Galambos & Simonelli (2004, Theorem 3.5)


  34. ^ ab Bryc (1995, p. 35)


  35. ^ ab Lukacs & King (1954)


  36. ^ Quine, M.P. (1993). "On three characterisations of the normal distribution". Probability and Mathematical Statistics. 14 (2): 257–263.


  37. ^ UIUC, Lecture 21. The Multivariate Normal Distribution, 21.6:"Individually Gaussian Versus Jointly Gaussian".


  38. ^ Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", The American Statistician, volume 36, number 4 November 1982, pages 372–373


  39. ^ "Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions". Allisons.org. 2007-12-05. Retrieved 2017-03-03.


  40. ^ Jordan, Michael I. (February 8, 2010). "Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution" (PDF).


  41. ^ Amari & Nagaoka (2000)


  42. ^ Weisstein, Eric W. "Normal Product Distribution". MathWorld. wolfram.com.


  43. ^ Lukacs, Eugene (1942). "A Characterization of the Normal Distribution". The Annals of Mathematical Statistics. Institute of Mathematical Statistics. 13 (1): 91–3. doi:10.1214/aoms/1177731647. ISSN 0003-4851. JSTOR 2236166. (Registration required (help)).


  44. ^ Basu, D.; Laha, R. G. (1954). "On Some Characterizations of the Normal Distribution". Sankhyā. Indian Statistical Institute. 13 (4): 359–62. ISSN 0036-4452. JSTOR 25048183. (Registration required (help)).


  45. ^ Lehmann, E. L. (1997). Testing Statistical Hypotheses (2nd ed.). Springer. p. 199. ISBN 0-387-94919-4.


  46. ^ John, S (1982). "The three parameter two-piece normal family of distributions and its fitting". Communications in statistics - Theory and Methods. 11 (8): 879–885. doi:10.1080/03610928208828279.


  47. ^ ab Krishnamoorthy (2006, p. 127)


  48. ^ Krishnamoorthy (2006, p. 130)


  49. ^ Krishnamoorthy (2006, p. 133)


  50. ^ Huxley (1932)


  51. ^ Jaynes, Edwin T. (2003). Probability Theory: The Logic of Science. Cambridge University Press. pp. 592–593.


  52. ^ Oosterbaan, Roland J. (1994). "Chapter 6: Frequency and Regression Analysis of Hydrologic Data". In Ritzema, Henk P. Drainage Principles and Applications, Publication 16 (PDF) (second revised ed.). Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175–224. ISBN 90-70754-33-9.


  53. ^ Wichura, Michael J. (1988). "Algorithm AS241: The Percentage Points of the Normal Distribution". Applied Statistics. Blackwell Publishing. 37 (3): 477–84. doi:10.2307/2347330. JSTOR 2347330. (Registration required (help)).


  54. ^ Johnson, Kotz & Balakrishnan (1995, Equation (26.48))


  55. ^ Kinderman & Monahan (1977)


  56. ^ Leva (1992)


  57. ^ Marsaglia & Tsang (2000)


  58. ^ Karney (2016)


  59. ^ Monahan (1985, section 2)


  60. ^ Wallace (1996)


  61. ^ Johnson, Kotz & Balakrishnan (1994, p. 85)


  62. ^ Le Cam & Lo Yang (2000, p. 74)


  63. ^ De Moivre, Abraham (1733), Corollary I – see Walker (1985, p. 77)


  64. ^ Stigler (1986, p. 76)


  65. ^ Gauss (1809, section 177)


  66. ^ Gauss (1809, section 179)


  67. ^ Laplace (1774, Problem III)


  68. ^ Pearson (1905, p. 189)


  69. ^ Stigler (1986, p. 144)


  70. ^ Stigler (1978, p. 243)


  71. ^ Stigler (1978, p. 244)


  72. ^ Maxwell (1860, p. 23)


  73. ^ Jaynes, Edwin J.; Probability Theory: The Logic of Science, Ch 7


  74. ^ Peirce, Charles S. (c. 1909 MS), Collected Papers v. 6, paragraph 327


  75. ^ Kruskal & Stigler (1997)


  76. ^ "Earliest uses… (entry STANDARD NORMAL CURVE)".




Sources


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External links








  • Hazewinkel, Michiel, ed. (2001) [1994], "Normal distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4


  • Normal Distribution Video Tutorial Part 1-2 on YouTube

  • Normal distribution calculator


  • An 8-foot-tall (2.4 m) Probability Machine (named Sir Francis) comparing stock market returns to the randomness of the beans dropping through the quincunx pattern. on YouTube Link originating from Index Fund Advisors









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