Normal distribution

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https://en.wikipedia.org/wiki/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	${\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$
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	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle 0}$
Entropy	${\displaystyle {\frac {1}{2}}\log(2\pi \sigma ^{2})+{\frac {1}{2}}}$
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	${\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\right\}}$

In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

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

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 statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. 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.

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

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:

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

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

Authors differ on which normal distribution should be called the "standard" one. Carl Friedrich Gauss, for example, defined the standard normal as having a variance of ${\displaystyle \sigma ^{2}=1/2}$. That is:

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

On the other hand, Stephen Stigler goes even further, defining the standard normal as having a variance of ${\displaystyle \sigma ^{2}=1/(2\pi )}$:

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

General normal distribution

Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor ${\displaystyle \sigma }$ (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 ${\displaystyle \phi }$ (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 ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$. 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, ${\displaystyle 1/\sigma ^{2}}$. 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.

Cumulative distribution function

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:

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

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 CDF, ${\displaystyle Q(x)=1-\Phi (x)}$, is often called the Q-function, especially in engineering texts. 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.

The graph of the standard normal CDF ${\displaystyle \Phi }$ 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 CDF 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 CDF for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion.

A quick approximation to the standard normal distribution's CDF can be found by using a taylor series approximation:

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

Standard deviation and coverage

Further information: Interval estimation and Coverage probability

 

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 ${\displaystyle \mu -n\sigma }$ 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 figures, 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 ${\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}$.

Quantile function

Further information: Quantile function § Normal distribution

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. Note that 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 ${\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}$

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. 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.

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

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.
• 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.

Moments

See also: List of integrals of Gaussian functions

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:

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

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	${\displaystyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}$	${\displaystyle 105\sigma ^{8}}$

If the random variable ${\displaystyle X}$ is Normally distributed with mean ${\displaystyle \mu }$ and finite non-zero variance ${\displaystyle \sigma ^{2}}$, then for ${\displaystyle 0<\delta <1}$ the expected value of the reciprocal for the absolute value of ${\displaystyle X}$ is

${\displaystyle {\begin{aligned}\operatorname {E} \left[{\frac {1}{\vert X\vert ^{\delta }}}\right]&\leq 2^{\frac {(1-\delta )}{2}}{\frac {\Gamma \left({\frac {1-\delta }{2}}\right)}{\sigma ^{\delta }{\sqrt {2\pi }}}}.\end{aligned}}}$

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 ${\displaystyle \sigma }$.

Fourier transform and characteristic function

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

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

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

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

Moment and cumulant generating functions

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

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

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

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

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

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 Dirac's "delta function" ${\displaystyle \delta }$ translated by the mean ${\displaystyle \mu }$, that is ${\displaystyle f(x)=\delta (x-\mu ).}$ Its CDF 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 mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, the normal distribution ${\displaystyle N(\mu ,\sigma ^{2})}$ is the one with maximum entropy. If ${\displaystyle X}$ is a continuous random variable with probability density ${\displaystyle f(x)}$, then the entropy of ${\displaystyle X}$ is defined as

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

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 variance, by using variational calculus. A function with two Lagrange multipliers is defined:

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

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

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:

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

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

${\displaystyle f(x)=e^{-\lambda _{0}-1-\lambda (x-\mu )^{2}}}$

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

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

The entropy of a normal distribution is equal to

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

Other properties

1. If the characteristic function ${\displaystyle \phi _{X}}$ of some random variable ${\displaystyle X}$ is of the form ${\displaystyle \phi _{X}(t)=\exp ^{Q(t)}}$, where ${\displaystyle Q(t)}$ is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that ${\displaystyle Q}$ can be at most a quadratic polynomial, and therefore ${\displaystyle X}$ is a normal random variable. 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are jointly normal and uncorrelated, then they are independent. The requirement that ${\displaystyle X}$ and ${\displaystyle Y}$ should be jointly normal is essential; without it the property does not hold. For non-normal random variables uncorrelatedness does not imply independence.
3. The Kullback–Leibler divergence of one normal distribution ${\displaystyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}$ from another ${\displaystyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}$ is given by:

   ${\displaystyle D_{\mathrm {KL} }(X_{1}\,\|\,X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}$

   The Hellinger distance between the same distributions is equal to

   ${\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}e^{-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}}$

4. The Fisher information matrix for a normal distribution is diagonal and takes the form
   ${\displaystyle {\mathcal {I}}={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&0\\0&{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}$
5. The conjugate prior of the mean of a normal distribution is another normal distribution. Specifically, if ${\displaystyle x_{1},\ldots ,x_{n}}$ are iid ${\displaystyle \sim N(\mu ,\sigma ^{2})}$ and the prior is ${\displaystyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}$, then the posterior distribution for the estimator of ${\displaystyle \mu }$ will be
   ${\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}$
6. The family of normal distributions not only forms an exponential family (EF), but in fact forms a natural exponential family (NEF) with quadratic variance function (NEF-QVF). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.
7. In information geometry, the family of normal distributions forms a statistical manifold with constant curvature ${\displaystyle -1}$. The same family is flat with respect to the (±1)-connections ∇${\displaystyle ^{(e)}}$ and ∇${\displaystyle ^{(m)}}$.