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[6] once defined the standard normal as

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

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

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 ${\displaystyle X}$.

The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter ${\displaystyle \phi }$ (phi).[7] 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 ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$.[8] Thus when a random variable ${\displaystyle X}$ is normally distributed with mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$, one may write

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

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, ${\displaystyle 1/\sigma ^{2}}$.[9] 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 ${\displaystyle \tau ^{\prime }=1/\sigma }$ might be defined as the precision, in which case the expression of the normal distribution becomes

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

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.

Normal distributions form an exponential family with natural parameters ${\displaystyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}$ and ${\displaystyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}$, and natural statistics x and x². The dual expectation parameters for normal distribution are η₁ = μ and η₂ = μ² + σ².

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

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

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 special functions. However, many numerical approximations are known; see below for more.

The two functions are closely related, namely

${\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}$

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

${\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$

The complement of the standard normal cumulative distribution function, ${\displaystyle Q(x)=1-\Phi (x)}$, is often called the Q-function, especially in engineering texts.[10][11] 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.[12]

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.[13]

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

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

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:

${\displaystyle \Phi ^{(0)}(x_{0})={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt}$

${\displaystyle \Phi ^{(1)}(x_{0})={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}}$

${\displaystyle \Phi ^{(n)}(x_{0})=-(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0}))}$, for all n ≥ 2.

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 ${\displaystyle \Phi (x_{0})}$

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

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, ${\displaystyle \Phi ({\text{desired}})}$. If ${\displaystyle x_{0}}$ is a good beginning estimate, convergence should be rapid with only a small number of iterations needed.

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.[5] 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:

${\displaystyle n}$	${\displaystyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}$	${\displaystyle {\text{i.e. }}1-p}$	${\displaystyle {\text{or }}1{\text{ in }}p}$	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 ${\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}$.

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:

${\displaystyle \Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}$

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

${\displaystyle F^{-1}(p)=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}$

The quantile ${\displaystyle \Phi ^{-1}(p)}$ of the standard normal distribution is commonly denoted as ${\displaystyle z_{p}}$. These values are used in hypothesis testing, construction of confidence intervals and Q–Q plots. 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 tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions. 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 ${\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}$

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.[18]
• 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 ${\displaystyle f^{\prime }(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}$
• 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 ${\displaystyle x=\mu +\sigma .}$[18]
• Its density is log-concave.[18]
• Its density is infinitely differentiable, indeed supersmooth of order 2.[19]

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

• Its first derivative is ${\displaystyle \varphi ^{\prime }(x)=-x\varphi (x).}$
• Its second derivative is ${\displaystyle \varphi ^{\prime \prime }(x)=(x^{2}-1)\varphi (x)}$
• 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.[20]
• 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:[21]

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

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 ${\displaystyle U.}$

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

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 inverse Mills ratio. Note that above, density ${\displaystyle f}$ of ${\displaystyle X}$ is used instead of standard normal density as in inverse Mills ratio, so here we have ${\displaystyle \sigma ^{2}}$ instead of ${\displaystyle \sigma }$.

The Fourier transform of a normal density ${\displaystyle f}$ with mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$ is[22]

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

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 ${\displaystyle t}$.[23] The relation between both is:

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

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

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

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

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

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

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 ${\displaystyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]<\infty }$.

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

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

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.[24] To see this, let ${\displaystyle X}$ be a continuous random variable with probability density ${\displaystyle f(x)}$. The entropy of ${\displaystyle X}$ is defined as[25][26][27]

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

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 variational calculus. A function with three Lagrange multipliers is defined:

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

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 ${\displaystyle \mu }$.

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

Comparison of probability density functions, ${\displaystyle p(k)}$ for the sum of ${\displaystyle n}$ fair 6-sided dice to show their convergence to a normal distribution with increasing ${\displaystyle na}$, 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 ${\displaystyle X_{1},\ldots ,X_{n}}$ are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance ${\displaystyle \sigma ^{2}}$ and ${\displaystyle Z}$ is their mean scaled by ${\displaystyle {\sqrt {n}}}$

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

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

The theorem can be extended to variables ${\displaystyle (X_{i})}$ 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 ${\displaystyle B(n,p)}$ is approximately normal with mean ${\displaystyle np}$ and variance ${\displaystyle np(1-p)}$ for large ${\displaystyle n}$ and for ${\displaystyle p}$ not too close to 0 or 1.
• The Poisson distribution with parameter ${\displaystyle \lambda }$ is approximately normal with mean ${\displaystyle \lambda }$ and variance ${\displaystyle \lambda }$, for large values of ${\displaystyle \lambda }$.[34]
• The chi-squared distribution ${\displaystyle \chi ^{2}(k)}$ is approximately normal with mean ${\displaystyle k}$ and variance ${\displaystyle 2k}$, for large ${\displaystyle k}$.
• The Student's t-distribution ${\displaystyle t(\nu )}$ is approximately normal with mean 0 and variance 1 when ${\displaystyle \nu }$ 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.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise. See AWGN.

a: Probability density of a function ${\displaystyle \cos x^{2}}$ of a normal variable ${\displaystyle x}$ with ${\displaystyle \mu =-2}$ and ${\displaystyle \sigma =3}$. b: Probability density of a function ${\displaystyle x^{y}}$ of two normal variables ${\displaystyle x}$ and ${\displaystyle y}$, where ${\displaystyle \mu _{x}=1}$, ${\displaystyle \mu _{y}=2}$, ${\displaystyle \sigma _{x}=0.1}$, ${\displaystyle \sigma _{y}=0.2}$, and ${\displaystyle \rho _{xy}=0.8}$. c: Heat map of the joint probability density of two functions of two correlated normal variables ${\displaystyle x}$ and ${\displaystyle y}$, where ${\displaystyle \mu _{x}=-2}$, ${\displaystyle \mu _{y}=5}$, ${\displaystyle \sigma _{x}^{2}=10}$, ${\displaystyle \sigma _{y}^{2}=20}$, and ${\displaystyle \rho _{xy}=0.495}$. d: Probability density of a function ${\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }$ of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.[35]

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing[35] (Matlab code). In the following sections we look at some special cases.

If ${\displaystyle X}$ is distributed normally with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, then

• ${\displaystyle aX+b}$, for any real numbers ${\displaystyle a}$ and ${\displaystyle b}$, is also normally distributed, with mean ${\displaystyle a\mu +b}$ and standard deviation ${\displaystyle |a|\sigma }$. That is, the family of normal distributions is closed under linear transformations.
• The exponential of ${\displaystyle X}$ is distributed log-normally: ${\displaystyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}$.
• The absolute value of ${\displaystyle X}$ has folded normal distribution: ${\displaystyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}$. If ${\displaystyle \mu =0}$ this is known as the half-normal distribution.
• The absolute value of normalized residuals, ${\displaystyle |X-\mu |/\sigma }$, has chi distribution with one degree of freedom: ${\displaystyle |X-\mu |/\sigma \sim \chi _{1}}$.
• The square of ${\displaystyle X/\sigma }$ has the noncentral chi-squared distribution with one degree of freedom: ${\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}$. If ${\displaystyle \mu =0}$, the distribution is called simply chi-squared.
• The log-likelihood of a normal variable ${\displaystyle x}$ is simply the log of its probability density function:
  $${\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}$$

  
  Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
• The distribution of the variable ${\displaystyle X}$ restricted to an interval ${\displaystyle [a,b]}$ is called the truncated normal distribution.
• ${\displaystyle (X-\mu )^{-2}}$ has a Lévy distribution with location 0 and scale ${\displaystyle \sigma ^{-2}}$.