Normal distribution

Normal distribution

Probability density function The red curve is the standard normal distribution.
Cumulative distribution function
Notation	${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$
Parameters	${\displaystyle \mu \in \mathbb {R} }$ = mean (location) ${\displaystyle \sigma ^{2}\in \mathbb {R} _{>0}}$ = variance (squared scale)
Support	${\displaystyle x\in \mathbb {R} }$
PDF	${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$
CDF	${\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$
Quantile	${\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle \sigma ^{2}}$
MAD	${\displaystyle \sigma {\sqrt {2/\pi }}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 0}$
Entropy	${\displaystyle {\frac {1}{2}}\log(2\pi e\sigma ^{2})}$
MGF	${\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}$
CF	${\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}$
Fisher information	${\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}}$ ${\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}$
Kullback–Leibler divergence	${\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}}$
Expected shortfall	${\displaystyle \mu -\sigma {\frac {{\frac {1}{\sqrt {2\pi }}}e^{\frac {-\left(q_{p}\left({\frac {X-\mu }{\sigma }}\right)\right)^{2}}{2}}}{1-p}}}$[1]

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v t e

In

The parameter ${\displaystyle \mu }$ is the mean or expectation of the distribution (and also its median and mode), while the parameter ${\displaystyle \sigma }$ is its standard deviation. The variance of the distribution is ${\displaystyle \sigma ^{2}}$. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Normal distributions are important in

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares^[5] parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.

A normal distribution is sometimes informally called a bell curve.^[6] However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). For other names, see Naming.

The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.

Definitions

Standard normal distribution

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

${\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}.}$

The variable ${\displaystyle z}$ has a mean of 0 and a variance and standard deviation of 1. The density ${\displaystyle \varphi (z)}$ has its peak ${\displaystyle 1/{\sqrt {2\pi }}}$ at ${\displaystyle z=0}$ and inflection points at ${\displaystyle z=+1}$ and ${\displaystyle z=-1}$.

Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss, for example, once defined the standard normal as

${\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}$

which has a variance of 1/2, and Stephen Stigler^[7] once defined the standard normal as

${\displaystyle \varphi (z)=e^{-\pi z^{2}},}$

which has a simple functional form and a variance of ${\displaystyle \sigma ^{2}=1/(2\pi ).}$

General normal distribution

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

${\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)}$

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

If ${\displaystyle Z}$ is a standard normal deviate, then ${\displaystyle X=\sigma Z+\mu }$ will have a normal distribution with expected value ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$. This is equivalent to saying that the standard normal distribution ${\displaystyle Z}$ can be scaled/stretched by a factor of ${\displaystyle \sigma }$ and shifted by ${\displaystyle \mu }$ to yield a different normal distribution, called ${\displaystyle X}$. Conversely, if ${\displaystyle X}$ is a normal deviate with parameters ${\displaystyle \mu }$ and ${\displaystyle \sigma ^{2}}$, then this ${\displaystyle X}$ distribution can be re-scaled and shifted via the formula ${\displaystyle Z=(X-\mu )/\sigma }$ to convert it to the standard normal distribution. This variate is also called the standardized form of ${\displaystyle X}$.

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 ${\displaystyle \phi }$ (

The alternative form of the Greek letter phi, ${\displaystyle \varphi }$, is also used quite often.

The normal distribution is often referred to as ${\displaystyle N(\mu ,\sigma ^{2})}$ or ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$.[9] Thus when a random variable ${\displaystyle X}$ is normally distributed with mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$, one may write

${\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}$

Alternative parameterizations

Some authors advocate using the precision ${\displaystyle \tau }$ as the parameter defining the width of the distribution, instead of the deviation ${\displaystyle \sigma }$ or the variance ${\displaystyle \sigma ^{2}}$. The precision is normally defined as the reciprocal of the variance, ${\displaystyle 1/\sigma ^{2}}$.^[10] The formula for the distribution then becomes

${\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.}$

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

Alternatively, the reciprocal of the standard deviation ${\displaystyle \tau '=1/\sigma }$ might be defined as the precision, in which case the expression of the normal distribution becomes

${\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.}$

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.

Normal distributions form an

${\displaystyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}$

${\displaystyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}$

Cumulative distribution function

The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter ${\displaystyle \Phi }$ (

${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt}$

Error Function

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

${\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}$

These integrals cannot be expressed in terms of elementary functions, and are often said to be

The complement of the standard normal cumulative distribution function, ${\displaystyle Q(x)=1-\Phi (x)}$, is often called the Q-function, especially in engineering texts.^[11][12] It gives the probability that the value of a standard normal random variable ${\displaystyle X}$ will exceed ${\displaystyle x}$: ${\displaystyle P(X>x)}$. Other definitions of the ${\displaystyle Q}$-function, all of which are simple transformations of ${\displaystyle \Phi }$, are also used occasionally.^[13]

The graph of the standard normal cumulative distribution function ${\displaystyle \Phi }$ has 2-fold rotational symmetry around the point (0,1/2); that is, ${\displaystyle \Phi (-x)=1-\Phi (x)}$. Its antiderivative (indefinite integral) can be expressed as follows:

${\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.}$

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

${\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]}$

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

An asymptotic expansion of the cumulative distribution function for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion.^[14]

A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation:

${\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}}$

Recursive computation with Taylor series expansion

The recursive nature of the ${\displaystyle e^{ax^{2}}}$family of derivatives may be used to easily construct a rapidly converging Taylor series expansion using recursive entries about any point of known value of the distribution,${\displaystyle \Phi (x_{0})}$:

${\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}$

where:

${\displaystyle {\begin{aligned}\Phi ^{(0)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt\\\Phi ^{(1)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}\\\Phi ^{(n)}(x_{0})&=-\left(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0})\right),&n\geq 2\end{aligned}}}$

Using the Taylor series and Newton's method for the inverse function

An application for the above Taylor series expansion is to use Newton's method to reverse the computation. That is, if we have a value for the cumulative distribution function, ${\displaystyle \Phi (x)}$, but do not know the x needed to obtain the ${\displaystyle \Phi (x)}$, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of ${\displaystyle \Phi (x)}$, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution.

To solve, select a known approximate solution, ${\displaystyle x_{0}}$, to the desired ${\displaystyle \Phi (x)}$. ${\displaystyle x_{0}}$ may be a value from a distribution table, or an intelligent estimate followed by a computation of ${\displaystyle \Phi (x_{0})}$ using any desired means to compute. Use this value of ${\displaystyle x_{0}}$ and the Taylor series expansion above to minimize computations.

Repeat the following process until the difference between the computed ${\displaystyle \Phi (x_{n})}$ and the desired ${\displaystyle \Phi }$, which we will call ${\displaystyle \Phi ({\text{desired}})}$, is below a chosen acceptably small error, such as 10⁻⁵, 10⁻¹⁵, etc.:

$${\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi '(x_{n})}}}$$

where

${\displaystyle \Phi (x,x_{0},\Phi (x_{0}))}$ is the ${\displaystyle \Phi (x)}$ from a Taylor series solution using ${\displaystyle x_{0}}$ and ${\displaystyle \Phi (x_{0})}$

${\displaystyle \Phi '(x_{n})={\frac {1}{\sqrt {2\pi }}}e^{-x_{n}^{2}/2}}$

When the repeated computations converge to an error below the chosen acceptably small value, x will be the value needed to obtain a ${\displaystyle \Phi (x)}$ of the desired value, ${\displaystyle \Phi ({\text{desired}})}$.

Standard deviation and coverage

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.^[6] 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 ${\displaystyle \mu -n\sigma }$ and ${\displaystyle \mu +n\sigma }$ is given by

${\displaystyle F(\mu +n\sigma )-F(\mu -n\sigma )=\Phi (n)-\Phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).}$

To 12 significant digits, the values for ${\displaystyle n=1,2,\ldots ,6}$ are:^{[citation needed]}

${\displaystyle n}$	${\displaystyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}$	${\displaystyle {\text{i.e. }}1-p}$	${\displaystyle {\text{or }}1{\text{ in }}p}$	OEIS
1	0.682689492137	0.317310507863	3 .15148718753	OEIS: A178647
2	0.954499736104	0.045500263896	21 .9778945080	OEIS: A110894
3	0.997300203937	0.002699796063	370 .398347345	OEIS: A270712
4	0.999936657516	0.000063342484	15787 .1927673
5	0.999999426697	0.000000573303	1744277 .89362
6	0.999999998027	0.000000001973	506797345 .897

For large ${\displaystyle n}$, one can use the approximation ${\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}$.

Quantile function

The

The quantile ${\displaystyle \Phi ^{-1}(p)}$ of the standard normal distribution is commonly denoted as ${\displaystyle z_{p}}$. These values are used in

. A normal random variable ${\displaystyle X}$ will exceed ${\displaystyle \mu +z_{p}\sigma }$ with probability ${\displaystyle 1-p}$, and will lie outside the interval ${\displaystyle \mu \pm z_{p}\sigma }$ with probability ${\displaystyle 2(1-p)}$. In particular, the quantile ${\displaystyle z_{0.975}}$ is

1.96

; therefore a normal random variable will lie outside the interval ${\displaystyle \mu \pm 1.96\sigma }$ in only 5% of cases.

The following table gives the quantile ${\displaystyle z_{p}}$ such that ${\displaystyle X}$ will lie in the range ${\displaystyle \mu \pm z_{p}\sigma }$ with a specified probability ${\displaystyle p}$. These values are useful to determine

The following table shows ${\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)}$, not ${\displaystyle \Phi ^{-1}(p)}$ as defined above.

${\displaystyle p}$	${\displaystyle z_{p}}$		${\displaystyle p}$	${\displaystyle z_{p}}$
0.80	1.281551565545		0.999	3.290526731492
0.90	1.644853626951		0.9999	3.890591886413
0.95	1.959963984540		0.99999	4.417173413469
0.98	2.326347874041		0.999999	4.891638475699
0.99	2.575829303549		0.9999999	5.326723886384
0.995	2.807033768344		0.99999999	5.730728868236
0.998	3.090232306168		0.999999999	6.109410204869

For small ${\displaystyle p}$, the quantile function has the useful asymptotic expansion ${\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}$^{[citation needed]}

Properties

The normal distribution is the only 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.^[16][17] 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.^[18][19]

The normal distribution is a subclass of the

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 ${\displaystyle f(x)}$ (mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma >0}$) has the following properties:

• It is symmetric around the point ${\displaystyle x=\mu ,}$ which is at the same time the mode, the median and the mean of the distribution.^[20]
• It is
  unimodal: its first derivative
  is positive for ${\displaystyle x<\mu ,}$ negative for ${\displaystyle x>\mu ,}$ and zero only at ${\displaystyle x=\mu .}$
• The area bounded by the curve and the ${\displaystyle x}$-axis is unity (i.e. equal to one).
• Its first derivative is ${\displaystyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}$
• Its second derivative is ${\displaystyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}$
• Its density has two inflection points (where the second derivative of ${\displaystyle f}$ is zero and changes sign), located one standard deviation away from the mean, namely at ${\displaystyle x=\mu -\sigma }$ and ${\displaystyle x=\mu +\sigma .}$^[20]
• Its density is log-concave.^[20]
• Its density is infinitely
  supersmooth of order 2.^[21]

• Its first derivative is ${\displaystyle \varphi '(x)=-x\varphi (x).}$
• Its second derivative is ${\displaystyle \varphi ''(x)=(x^{2}-1)\varphi (x)}$
• More generally, its nth derivative is ${\displaystyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}$ where ${\displaystyle \operatorname {He} _{n}(x)}$ is the nth (probabilist)
  Hermite polynomial.^[22]
• The probability that a normally distributed variable ${\displaystyle X}$ with known ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ is in a particular set, can be calculated by using the fact that the fraction ${\displaystyle Z=(X-\mu )/\sigma }$ has a standard normal distribution.

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

If ${\displaystyle X}$ has a normal distribution, the non-central moments exist and are finite for any ${\displaystyle p}$ whose real part is greater than −1. For any non-negative integer ${\displaystyle p}$, the plain central moments are:^[23]

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

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

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

${\displaystyle {\begin{aligned}\operatorname {E} \left[|X-\mu |^{p}\right]&=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{if }}p{\text{ is odd}}\\1&{\text{if }}p{\text{ is even}}\end{cases}}\\&=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}}$

The last formula is valid also for any non-integer ${\displaystyle p>-1.}$ When the mean ${\displaystyle \mu \neq 0,}$ the plain and absolute moments can be expressed in terms of confluent hypergeometric functions ${\displaystyle {}_{1}F_{1}}$ and ${\displaystyle U.}$^[24]

${\displaystyle {\begin{aligned}\operatorname {E} \left[X^{p}\right]&=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {E} \left[|X|^{p}\right]&=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}}$

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

Order	Non-central moment	Central moment
1	${\displaystyle \mu }$	${\displaystyle 0}$
2	${\displaystyle \mu ^{2}+\sigma ^{2}}$	${\displaystyle \sigma ^{2}}$
3	${\displaystyle \mu ^{3}+3\mu \sigma ^{2}}$	${\displaystyle 0}$
4	${\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}$	${\displaystyle 3\sigma ^{4}}$
5	${\displaystyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}}$	${\displaystyle 0}$
6	${\displaystyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}}$	${\displaystyle 15\sigma ^{6}}$
7	${\displaystyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}}$	${\displaystyle 0}$
8	${\displaystyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}$	${\displaystyle 105\sigma ^{8}}$

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

${\displaystyle \operatorname {E} \left[X\mid a<X<b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}}$

where ${\displaystyle f}$ and ${\displaystyle F}$ respectively are the density and the cumulative distribution function of ${\displaystyle X}$. For ${\displaystyle b=\infty }$ this is known as the

${\displaystyle f}$

${\displaystyle X}$

${\displaystyle \sigma ^{2}}$

${\displaystyle \sigma }$

Fourier transform and characteristic function

The Fourier transform of a normal density ${\displaystyle f}$ with mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$ is^[25]

${\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}}$

where ${\displaystyle i}$ is the imaginary unit. If the mean ${\displaystyle \mu =0}$, 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 ${\displaystyle 1/\sigma }$. In particular, the standard normal distribution ${\displaystyle \varphi }$ is an eigenfunction of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable ${\displaystyle X}$ is closely connected to the characteristic function ${\displaystyle \varphi _{X}(t)}$ of that variable, which is defined as the expected value of ${\displaystyle e^{itX}}$, as a function of the real variable ${\displaystyle t}$ (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable ${\displaystyle t}$.^[26] The relation between both is:

${\displaystyle \varphi _{X}(t)={\hat {f}}(-t)}$

Moment- and cumulant-generating functions

The

${\displaystyle X}$

${\displaystyle e^{tX}}$

${\displaystyle t}$

${\displaystyle f}$

${\displaystyle \mu }$

${\displaystyle \sigma }$

${\displaystyle M(t)=\operatorname {E} \left[e^{tX}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}}$

The

${\displaystyle g(t)=\ln M(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}}$

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

Some authors prefer to instead work with E[e^itX] = e^{iμt − σ2t2/2} and ln E[e^itX] = iμt − 1/2σ²t².

Stein operator and class

Within Stein's method the Stein operator and class of a random variable ${\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ are ${\displaystyle {\mathcal {A}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)}$ and ${\displaystyle {\mathcal {F}}}$ the class of all absolutely continuous functions ${\displaystyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]<\infty }$.

Zero-variance limit

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

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

${\displaystyle F(x)={\begin{cases}0&{\text{if }}x<\mu \\1&{\text{if }}x\geq \mu \end{cases}}}$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean ${\displaystyle \mu }$ and finite variance ${\displaystyle \sigma ^{2}}$, the normal distribution ${\displaystyle N(\mu ,\sigma ^{2})}$ is the one with maximum entropy.^[27] To see this, let ${\displaystyle X}$ be a

${\displaystyle f(x)}$. The entropy of ${\displaystyle X}$ is defined as

${\displaystyle H(X)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}$

where ${\displaystyle f(x)\log f(x)}$ is understood to be zero whenever ${\displaystyle f(x)=0}$. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using

${\displaystyle L=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.}$

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

${\displaystyle 0=\delta L=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx}$

Since this must hold for any small ${\displaystyle \delta f(x)}$, the factor multiplying ${\displaystyle \delta f(x)}$ must be zero, and solving for ${\displaystyle f(x)}$ yields:

${\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.}$

The Lagrange constraints that ${\displaystyle f(x)}$ is properly normalized and has the specified mean and variance are satisfied if and only if ${\displaystyle \lambda _{0}}$, ${\displaystyle \lambda _{1}}$, and ${\displaystyle \lambda _{2}}$ are chosen so that

${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}$

The entropy of a normal distribution ${\displaystyle X\sim N(\mu ,\sigma ^{2})}$ is equal to

${\displaystyle H(X)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,}$

which is independent of the mean ${\displaystyle \mu }$.

Other properties

${\displaystyle -1}$. The same family is

${\displaystyle \nabla ^{(e)}}$

${\displaystyle \nabla ^{(m)}}$

If ${\displaystyle X_{1},\dots ,X_{n}}$ are distributed according to ${\displaystyle N(0,\sigma ^{2})}$, then ${\displaystyle E[\max _{i}X_{i}]\leq \sigma {\sqrt {2\ln n}}}$. Note that there is no assumption of independence.^[37]