View source for Normal distribution

{{MSC|60E99}}

[[Category:Distribution theory]]

One of the most important probability distributions. The term "normal distribution" is due to K. Pearson (earlier names are [[Gauss law|Gauss law]] and [[Gauss–Laplace distribution|Gauss–Laplace distribution]]). It is used both in relation to probability distributions of random variables (cf. [[Random variable|Random variable]]) and in relation to the joint probability distribution (cf. [[Joint distribution|Joint distribution]]) of several random variables (that is, to distributions of finite-dimensional random vectors), as well as of random elements and stochastic processes (cf. [[Random element|Random element]]; [[Stochastic process|Stochastic process]]). The general definition of a normal distribution reduces to the one-dimensional case.

The probability distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674601.png" /> is called normal if it has probability density

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

The family of normal distributions (*) depends, as a rule, on the two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674604.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674605.png" /> is the mathematical expectation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674607.png" /> is the variance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674608.png" /> and the characteristic function has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n0674609.png" /></td> </tr></table>

The normal density curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746010.png" /> is symmetric about the ordinate passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746011.png" /> and has there its unique maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746012.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746013.png" /> decreases, the normal distribution curve becomes more and more pointed. A change in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746014.png" /> with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746015.png" /> does not change the shape of the curve and causes only a shift along the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746016.png" />-axis. The area under a normal density curve is 1. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746018.png" />, the corresponding distribution function is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746019.png" /></td> </tr></table>

In general, the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746020.png" /> of (*) can be computed by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746022.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746023.png" /> (and several of its derivatives) extensive tables have been compiled (see, for example, {{Cite|BS}}, {{Cite|T}}, and [[Probability integral|Probability integral]]). For a normal distribution the probability that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746025.png" /> and it decreases very rapidly with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746026.png" /> (see the Table).<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">k</td> <td colname="2" style="background-color:white;" colspan="1">probability</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1">0.31731</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">2</td> <td colname="2" style="background-color:white;" colspan="1">0.45500<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746027.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">3</td> <td colname="2" style="background-color:white;" colspan="1">0.26998<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746028.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">4</td> <td colname="2" style="background-color:white;" colspan="1">0.63342<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746029.png" /></td> </tr> </tbody> </table>
</td></tr> </table>

In many practical problems, when analyzing normal distributions one can, therefore, ignore the possibility of a deviation from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746030.png" /> in excess of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746031.png" /> — the three-sigma rule; the corresponding probability, as is clear from the Table, is less than 0.003. The quartile deviation for a normal distribution is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746032.png" />.

Normal distributions occur in a large number of applications. There are some noteable attempts at explaining this fact. A theoretical basis for the exceptional role of the normal distribution is given by the [[Limit theorems|limit theorems]] of probability theory (see also [[Laplace theorem|Laplace theorem]]; [[Lyapunov theorem|Lyapunov theorem]]). Qualitatively, the result can be stated in the following manner: A normal distribution is a good approximation whenever the relevant random variable is the sum of a large number of independent random variables the largest of which is small in comparison with the whole sum (see [[Central limit theorem|Central limit theorem]]).

A normal distribution can also appear as an exact solution of certain problems (within the framework of an accepted mathematical model of the phenomenon). This is so in the theory of random processes (in one of the basic models of Brownian motion). Classic examples of a normal distribution arising as an exact one are due to C.F. Gauss (the law of distribution of errors of observation) and J. Maxwell (the law of distribution of velocities of molecules) (see also [[Independence|Independence]]; [[Characterization theorems|Characterization theorems]]).

The distribution of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746034.png" />, or the joint distribution of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746035.png" />, is called normal (multivariate normal) if for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746036.png" /> the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746037.png" /> either has a normal distribution or is constant (as one sometimes says, has a normal distribution with variance zero). For random elements with values from some vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746038.png" /> this definition is retained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746039.png" /> is replaced by any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746040.png" /> of the adjoint space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746041.png" /> and the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746042.png" /> is replaced by a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746043.png" />. The joint distribution of several random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746044.png" /> has [[Characteristic function|characteristic function]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746045.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746046.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746047.png" /></td> </tr></table>

is a linear form,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746048.png" /></td> </tr></table>

is a non-negative definite quadratic form, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746049.png" /> is the [[Covariance matrix|covariance matrix]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746050.png" />. In the positive-definite case the corresponding normal distribution has the probability density

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746052.png" /> is the quadratic form inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746053.png" />, the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746054.png" /> are the mathematical expectations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746055.png" />, respectively, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746056.png" /></td> </tr></table>

is constant. The total number of parameters specifying the normal distribution is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746057.png" /></td> </tr></table>

and grows rapidly with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746058.png" /> (it is 2 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746059.png" />, 20 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746060.png" />, and 65 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746061.png" />). A multivariate normal distribution is the basic model of [[Multi-dimensional statistical analysis|multi-dimensional statistical analysis]]. It is also used in the theory of stochastic processes (where normal distributions in infinite-dimensional spaces are examined; see [[Random element|Random element]], and also [[Wiener measure|Wiener measure]]; [[Wiener process|Wiener process]]; [[Gaussian process|Gaussian process]]).

Of the important properties of normal distributions the following should be mentioned. The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746062.png" /> of two independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746064.png" /> having normal distributions also has a normal distribution; conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746065.png" /> has a normal distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746067.png" /> are independent, then the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746069.png" /> are normal (Cramér's theorem). This property has a certain "stability" : If the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746070.png" /> is "close" to normal, then so are the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746072.png" />. Some other important distributions are connected with normal ones (see [[Logarithmic normal distribution|Logarithmic normal distribution]]; [[Non-central chi-squared distribution|Non-central "chi-squared" distribution]]; [[Student distribution|Student distribution]]; [[Wishart distribution|Wishart distribution]]; [[Fisher z-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746073.png" />-distribution]]; [[Hotelling-T^2-distribution|Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067460/n06746074.png" />-distribution]]; [[Chi-squared distribution| Chi-squared distribution]]). For an approximate representation of distributions close to normal, series like [[Edgeworth series|Edgeworth series]] and [[Gram–Charlier series|Gram–Charlier series]] are widely used.

Concerning problems connected with estimators of parameters of normal distributions using results of observations see [[Unbiased estimator|Unbiased estimator]]. Concerning testing the hypothesis of normality see [[Non-parametric methods in statistics|Non-parametric methods in statistics]]. See also [[Probability graph paper|Probability graph paper]].

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|-
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|valign="top"|{{Ref|C}}|| H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
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====Comments====


====References====
{|
|valign="top"|{{Ref|JK}}|| N.L. Johnson, S. Kotz, "Distributions in statistics" , '''2. Continuous univariate distributions''' , Wiley (1970) {{MR|0270476}} {{MR|0270475}} {{ZBL|0213.21101}}
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