The parameter $\mu$ is the mean or expectation of the distribution (and also its median and mode), while the parameter $\sigma$ is its standard deviation. The variance of the distribution is $\sigma ^{2}$. 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.^{[2]}^{[3]} 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.^{[4]}
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^{[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 simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when $\mu =0$ and $\sigma =1$, and it is described by this probability density function (or density):
The variable $z$ has a mean of 0 and a variance and standard deviation of 1. The density $\varphi (z)$ has its peak $1/{\sqrt {2\pi }}$ at $z=0$ and inflection points at $z=+1$ and $z=-1$.
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
$\varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},$
which has a variance of 1/2, and Stephen Stigler^{[7]} once defined the standard normal as
$\varphi (z)=e^{-\pi z^{2}},$
which has a simple functional form and a variance of $\sigma ^{2}=1/(2\pi ).$
General normal distributionEdit
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor $\sigma$ (the standard deviation) and then translated by $\mu$ (the mean value):
The probability density must be scaled by $1/\sigma$ so that the integral is still 1.
If $Z$ is a standard normal deviate, then $X=\sigma Z+\mu$ will have a normal distribution with expected value $\mu$ and standard deviation $\sigma$. This is equivalent to saying that the "standard" normal distribution $Z$ can be scaled/stretched by a factor of $\sigma$ and shifted by $\mu$ to yield a different normal distribution, called $X$. Conversely, if $X$ is a normal deviate with parameters $\mu$ and $\sigma ^{2}$, then this $X$ distribution can be re-scaled and shifted via the formula $Z=(X-\mu )/\sigma$ to convert it to the "standard" normal distribution. This variate is also called the standardized form of $X$.
NotationEdit
The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter $\phi$ (phi).^{[8]} The alternative form of the Greek letter phi, $\varphi$, is also used quite often.
The normal distribution is often referred to as $N(\mu ,\sigma ^{2})$ or ${\mathcal {N}}(\mu ,\sigma ^{2})$.^{[9]} Thus when a random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, one may write
$X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).$
Alternative parameterizationsEdit
Some authors advocate using the precision$\tau$ as the parameter defining the width of the distribution, instead of the deviation $\sigma$ or the variance $\sigma ^{2}$. The precision is normally defined as the reciprocal of the variance, $1/\sigma ^{2}$.^{[10]} The formula for the distribution then becomes
This choice is claimed to have advantages in numerical computations when $\sigma$ 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 $\tau ^{\prime }=1/\sigma$ might be defined as the precision, in which case the expression of the normal distribution becomes
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$\textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}$ and $\textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}$, and natural statistics x and x^{2}. The dual expectation parameters for normal distribution are η_{1} = μ and η_{2} = μ^{2} + σ^{2}.
Cumulative distribution functionsEdit
The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter $\Phi$ (phi), is the integral
The related error function$\operatorname {erf} (x)$ gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range $[-x,x]$. That is:
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 complement of the standard normal CDF, $Q(x)=1-\Phi (x)$, 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 $X$ will exceed $x$: $P(X>x)$. Other definitions of the $Q$-function, all of which are simple transformations of $\Phi$, are also used occasionally.^{[13]}
The graph of the standard normal CDF $\Phi$ has 2-fold rotational symmetry around the point (0,1/2); that is, $\Phi (-x)=1-\Phi (x)$. Its antiderivative (indefinite integral) can be expressed as follows:
$\int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.$
The CDF of the standard normal distribution can be expanded by Integration by parts into a series:
Recursive computation with Taylor series expansionEdit
The recursive nature of the $e^{ax^{2}}$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,$\Phi (x_{0})$:
$\Phi ^{(n)}(x_{0})=-(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0}))$, for all n ≥ 2.
Using the Taylor series and Newton's method for the inverse functionEdit
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 CDF, $\Phi (x)$, but do not know the x needed to obtain the $\Phi (x)$, 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 $\Phi (x)$, 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, $x_{0}$, to the desired $\Phi (x)$. $x_{0}$ may be a value from a distribution table, or an intelligent estimate followed by a computation of $\Phi (x_{0})$ using any desired means to compute. Use this value of $x_{0}$ and the Taylor series expansion above to minimize computations.
Repeat the following process until the difference between the computed $\Phi (x_{n})$ and the desired $\Phi$, which we will call $\Phi ({\text{desired}})$, is below a chosen acceptably small error, such as 10^{−5}, 10^{−15}, etc.:
When the repeated computations converge to an error below the chosen acceptably small value, x will be the value needed to obtain a $\Phi (x)$ of the desired value, $\Phi ({\text{desired}})$. If $x_{0}$ is a good beginning estimate, convergence should be rapid with only a small number of iterations needed.^{[citation needed]}
Standard deviation and coverageEdit
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 $\mu -n\sigma$ and $\mu +n\sigma$ is given by
For large $n$, one can use the approximation $1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}$.
Quantile functionEdit
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:
The quantile$\Phi ^{-1}(p)$ of the standard normal distribution is commonly denoted as $z_{p}$. These values are used in hypothesis testing, construction of confidence intervals and Q–Q plots. A normal random variable $X$ will exceed $\mu +z_{p}\sigma$ with probability $1-p$, and will lie outside the interval $\mu \pm z_{p}\sigma$ with probability $2(1-p)$. In particular, the quantile $z_{0.975}$ is 1.96; therefore a normal random variable will lie outside the interval $\mu \pm 1.96\sigma$ in only 5% of cases.
The following table gives the quantile $z_{p}$ such that $X$ will lie in the range $\mu \pm z_{p}\sigma$ with a specified probability $p$. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions.^{[citation needed]} Note that the following table shows ${\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)$, not $\Phi ^{-1}(p)$ as defined above.
$p$
$z_{p}$
$p$
$z_{p}$
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 $p$, the quantile function has the useful asymptotic expansion$\Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).$^{[citation needed]}
PropertiesEdit
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.^{[15]}^{[16]} 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.^{[17]}^{[18]}
The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
The value of the normal distribution is practically zero when the value $x$ 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 derivativesEdit
The normal distribution with density $f(x)$ (mean $\mu$ and standard deviation $\sigma >0$) has the following properties:
It is symmetric around the point $x=\mu ,$ which is at the same time the mode, the median and the mean of the distribution.^{[19]}
It is unimodal: its first derivative is positive for $x<\mu ,$ negative for $x>\mu ,$ and zero only at $x=\mu .$
The area bounded by the curve and the $x$-axis is unity (i.e. equal to one).
Its first derivative is $f^{\prime }(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).$
Its density has two inflection points (where the second derivative of $f$ is zero and changes sign), located one standard deviation away from the mean, namely at $x=\mu -\sigma$ and $x=\mu +\sigma .$^{[19]}
Furthermore, the density $\varphi$ of the standard normal distribution (i.e. $\mu =0$ and $\sigma =1$) also has the following properties:
Its first derivative is $\varphi ^{\prime }(x)=-x\varphi (x).$
Its second derivative is $\varphi ^{\prime \prime }(x)=(x^{2}-1)\varphi (x)$
More generally, its nth derivative is $\varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),$ where $\operatorname {He} _{n}(x)$ is the nth (probabilist) Hermite polynomial.^{[21]}
The probability that a normally distributed variable $X$ with known $\mu$ and $\sigma$ is in a particular set, can be calculated by using the fact that the fraction $Z=(X-\mu )/\sigma$ has a standard normal distribution.
MomentsEdit
The plain and absolute moments of a variable $X$ are the expected values of $X^{p}$ and $|X|^{p}$, respectively. If the expected value $\mu$ of $X$ 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 $\ p$.
If $X$ has a normal distribution, the non-central moments exist and are finite for any $p$ whose real part is greater than −1. For any non-negative integer $p$, the plain central moments are:^{[22]}
$\operatorname {E} \left[(X-\mu )^{p}\right]={\begin{cases}0&{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&{\text{if }}p{\text{ is even.}}\end{cases}}$
Here $n!!$ denotes the double factorial, that is, the product of all numbers from $n$ to 1 that have the same parity as $n.$
The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer $p,$
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \neq 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric functions${}_{1}F_{1}$ and $U.$^{[citation needed]}
where $f$ and $F$ respectively are the density and the cumulative distribution function of $X$. For $b=\infty$ this is known as the inverse Mills ratio. Note that above, density $f$ of $X$ is used instead of standard normal density as in inverse Mills ratio, so here we have $\sigma ^{2}$ instead of $\sigma$.
Fourier transform and characteristic functionEdit
The Fourier transform of a normal density $f$ with mean $\mu$ and standard deviation $\sigma$ is^{[23]}
where $i$ is the imaginary unit. If the mean $\mu =0$, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and standard deviation $1/\sigma$. In particular, the standard normal distribution $\varphi$ is an eigenfunction of the Fourier transform.
In probability theory, the Fourier transform of the probability distribution of a real-valued random variable $X$ is closely connected to the characteristic function$\varphi _{X}(t)$ of that variable, which is defined as the expected value of $e^{itX}$, as a function of the real variable $t$ (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable $t$.^{[24]} The relation between both is:
$\varphi _{X}(t)={\hat {f}}(-t)$
Moment- and cumulant-generating functionsEdit
The moment generating function of a real random variable $X$ is the expected value of $e^{tX}$, as a function of the real parameter $t$. For a normal distribution with density $f$, mean $\mu$ and deviation $\sigma$, the moment generating function exists and is equal to
Since this is a quadratic polynomial in $t$, only the first two cumulants are nonzero, namely the mean $\mu$ and the variance $\sigma ^{2}$.
Stein operator and classEdit
Within Stein's method the Stein operator and class of a random variable $X\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ are ${\mathcal {A}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)$ and ${\mathcal {F}}$ the class of all absolutely continuous functions $f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]<\infty$.
Zero-variance limitEdit
In the limit when $\sigma$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\neq \mu$, but grows without limit if $x=\mu$, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when $\sigma =0$.
However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function"$\delta$ translated by the mean $\mu$, that is $f(x)=\delta (x-\mu ).$
Its CDF is then the Heaviside step function translated by the mean $\mu$, namely
Of all probability distributions over the reals with a specified mean $\mu$ and variance $\sigma ^{2}$, the normal distribution $N(\mu ,\sigma ^{2})$ is the one with maximum entropy.^{[25]} If $X$ is a continuous random variable with probability density$f(x)$, then the entropy of $X$ is defined as^{[26]}^{[27]}^{[28]}
where $f(x)\log f(x)$ is understood to be zero whenever $f(x)=0$. 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:
If the characteristic function $\phi _{X}$ of some random variable $X$ is of the form $\phi _{X}(t)=\exp Q(t)$, where $Q(t)$ is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that $Q$ can be at most a quadratic polynomial, and therefore $X$ is a normal random variable.^{[29]} The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero cumulants.
If $X$ and $Y$ are jointly normal and uncorrelated, then they are independent. The requirement that $X$ and $Y$ should be jointly normal is essential; without it the property does not hold.^{[30]}^{[31]}^{[proof]} For non-normal random variables uncorrelatedness does not imply independence.
The Kullback–Leibler divergence of one normal distribution $X_{1}\sim N(\mu _{1},\sigma _{1}^{2})$ from another $X_{2}\sim N(\mu _{2},\sigma _{2}^{2})$ is given by:^{[32]}
The conjugate prior of the mean of a normal distribution is another normal distribution.^{[33]} Specifically, if $x_{1},\ldots ,x_{n}$ are iid $\sim N(\mu ,\sigma ^{2})$ and the prior is $\mu \sim N(\mu _{0},\sigma _{0}^{2})$, then the posterior distribution for the estimator of $\mu$ will be
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.
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 $X_{1},\ldots ,X_{n}$ are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance $\sigma ^{2}$ and $Z$ is their
mean scaled by ${\sqrt {n}}$
Then, as $n$ increases, the probability distribution of $Z$ will tend to the normal distribution with zero mean and variance $\sigma ^{2}$.
The theorem can be extended to variables $(X_{i})$ 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 Poisson distribution with parameter $\lambda$ is approximately normal with mean $\lambda$ and variance $\lambda$, for large values of $\lambda$.^{[35]}
The chi-squared distribution$\chi ^{2}(k)$ is approximately normal with mean $k$ and variance $2k$, for large $k$.
The Student's t-distribution$t(\nu )$ is approximately normal with mean 0 and variance 1 when $\nu$ 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.
Operations and functions of normal variablesEdit
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^{[36]} (Matlab code). In the following sections we look at some special cases.
Operations on a single normal variableEdit
If $X$ is distributed normally with mean $\mu$ and variance $\sigma ^{2}$, then
$aX+b$, for any real numbers $a$ and $b$, is also normally distributed, with mean $a\mu +b$ and standard deviation $|a|\sigma$. That is, the family of normal distributions is closed under linear transformations.
The exponential of $X$ is distributed log-normally: $e^{X}\sim \ln(N(\mu ,\sigma ^{2}))$.
The absolute value of normalized residuals, $|X-\mu |/\sigma$, has chi distribution with one degree of freedom: $|X-\mu |/\sigma \sim \chi _{1}$.
The square of $X/\sigma$ 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 $\mu =0$, the distribution is called simply chi-squared.
$(X-\mu )^{-2}$ has a Lévy distribution with location 0 and scale $\sigma ^{-2}$.
Operations on two independent normal variablesEdit
If $X_{1}$ and $X_{2}$ are two independent normal random variables, with means $\mu _{1}$, $\mu _{2}$ and standard deviations $\sigma _{1}$, $\sigma _{2}$, then their sum $X_{1}+X_{2}$ will also be normally distributed,^{[proof]} with mean $\mu _{1}+\mu _{2}$ and variance $\sigma _{1}^{2}+\sigma _{2}^{2}$.
In particular, if $X$ and $Y$ are independent normal deviates with zero mean and variance $\sigma ^{2}$, then $X+Y$ and $X-Y$ are also independent and normally distributed, with zero mean and variance $2\sigma ^{2}$. This is a special case of the polarization identity.^{[37]}
If $X_{1}$, $X_{2}$ are two independent normal deviates with mean $\mu$ and deviation $\sigma$, and $a$, $b$ are arbitrary real numbers, then the variable
is also normally distributed with mean $\mu$ and deviation $\sigma$. It follows that the normal distribution is stable (with exponent $\alpha =2$).
If $X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})$, $k\in \{0,1\}$ are normal distributions, then their normalized geometric mean${\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }$ is a normal distribution ${\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})$ with $m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}$ and $\sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}$ (see here for a visualization).
Operations on two independent standard normal variablesEdit
If $X_{1}$ and $X_{2}$ are two independent standard normal random variables with mean 0 and variance 1, then
Their sum and difference is distributed normally with mean zero and variance two: $X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)$.
Operations on multiple independent normal variablesEdit
Any linear combination of independent normal deviates is a normal deviate.
If $X_{1},X_{2},\ldots ,X_{n}$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with $n$ degrees of freedom
$X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.$
If $X_{1},X_{2},\ldots ,X_{n}$ are independent normally distributed random variables with means $\mu$ and variances $\sigma ^{2}$, then their sample mean is independent from the sample standard deviation,^{[39]} which can be demonstrated using Basu's theorem or Cochran's theorem.^{[40]} The ratio of these two quantities will have the Student's t-distribution with $n-1$ degrees of freedom:
If $X_{1},X_{2},\ldots ,X_{n}$, $Y_{1},Y_{2},\ldots ,Y_{m}$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom:^{[41]}
Operations on multiple correlated normal variablesEdit
A quadratic form of a normal vector, i.e. a quadratic function ${\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}$ of multiple independent or correlated normal variables, is a generalized chi-square variable.
Operations on the density functionEdit
The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.
Infinite divisibility and Cramér's theoremEdit
For any positive integer ${\text{n}}$, any normal distribution with mean $\mu$ and variance $\sigma ^{2}$ is the distribution of the sum of ${\text{n}}$ independent normal deviates, each with mean ${\frac {\mu }{n}}$ and variance ${\frac {\sigma ^{2}}{n}}$. This property is called infinite divisibility.^{[42]}
Conversely, if $X_{1}$ and $X_{2}$ are independent random variables and their sum $X_{1}+X_{2}$ has a normal distribution, then both $X_{1}$ and $X_{2}$ must be normal deviates.^{[43]}
This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.^{[29]}
Bernstein's theoremEdit
Bernstein's theorem states that if $X$ and $Y$ are independent and $X+Y$ and $X-Y$ are also independent, then both X and Y must necessarily have normal distributions.^{[44]}^{[45]}
More generally, if $X_{1},\ldots ,X_{n}$ are independent random variables, then two distinct linear combinations ${\textstyle \sum {a_{k}X_{k}}}$ and ${\textstyle \sum {b_{k}X_{k}}}$will be independent if and only if all $X_{k}$ are normal and ${\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}}$, where $\sigma _{k}^{2}$ denotes the variance of $X_{k}$.^{[44]}
ExtensionsEdit
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R^{k} is multivariate-normally distributed if any linear combination of its components Σ^{k} _{j=1}a_{j} X_{j} has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ C^{k} is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert spaceH, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element h ∈ H is said to be normal if for any constant a ∈ H the scalar product(a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.
One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.
Statistical inferenceEdit
Estimation of parametersEdit
It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample $(x_{1},\ldots ,x_{n})$ from a normal ${\mathcal {N}}(\mu ,\sigma ^{2})$ population we would like to learn the approximate values of parameters $\mu$ and $\sigma ^{2}$. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:
Taking derivatives with respect to $\mu$ and $\sigma ^{2}$ and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Estimator $\textstyle {\hat {\mu }}$ is called the sample mean, since it is the arithmetic mean of all observations. The statistic $\textstyle {\overline {x}}$ is complete and sufficient for $\mu$, and therefore by the Lehmann–Scheffé theorem, $\textstyle {\hat {\mu }}$ is the uniformly minimum variance unbiased (UMVU) estimator.^{[47]} In finite samples it is distributed normally:
The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix$\textstyle {\mathcal {I}}^{-1}$. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of $\textstyle {\hat {\mu }}$ is proportional to $\textstyle 1/{\sqrt {n}}$, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.
The estimator $\textstyle {\hat {\sigma }}^{2}$ is called the sample variance, since it is the variance of the sample ($(x_{1},\ldots ,x_{n})$). In practice, another estimator is often used instead of the $\textstyle {\hat {\sigma }}^{2}$. This other estimator is denoted $s^{2}$, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root $s$ is called the sample standard deviation. The estimator $s^{2}$ differs from $\textstyle {\hat {\sigma }}^{2}$ by having (n − 1) instead of n in the denominator (the so-called Bessel's correction):
The difference between $s^{2}$ and $\textstyle {\hat {\sigma }}^{2}$ becomes negligibly small for large n's. In finite samples however, the motivation behind the use of $s^{2}$ is that it is an unbiased estimator of the underlying parameter $\sigma ^{2}$, whereas $\textstyle {\hat {\sigma }}^{2}$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^{2}$ is uniformly minimum variance unbiased (UMVU),^{[47]} which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle {\hat {\sigma }}^{2}$ is "better" than the $s^{2}$ in terms of the mean squared error (MSE) criterion. In finite samples both $s^{2}$ and $\textstyle {\hat {\sigma }}^{2}$ have scaled chi-squared distribution with (n − 1) degrees of freedom:
The first of these expressions shows that the variance of $s^{2}$ is equal to $2\sigma ^{4}/(n-1)$, which is slightly greater than the σσ-element of the inverse Fisher information matrix $\textstyle {\mathcal {I}}^{-1}$. Thus, $s^{2}$ is not an efficient estimator for $\sigma ^{2}$, and moreover, since $s^{2}$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma ^{2}$ does not exist.
Applying the asymptotic theory, both estimators $s^{2}$ and $\textstyle {\hat {\sigma }}^{2}$ are consistent, that is they converge in probability to $\sigma ^{2}$ as the sample size $n\rightarrow \infty$. The two estimators are also both asymptotically normal:
In particular, both estimators are asymptotically efficient for $\sigma ^{2}$.
Confidence intervalsEdit
By Cochran's theorem, for normal distributions the sample mean $\textstyle {\hat {\mu }}$ and the sample variance s^{2} are independent, which means there can be no gain in considering their joint distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle {\hat {\mu }}$ and s can be employed to construct the so-called t-statistic:
This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;^{[48]} similarly, inverting the χ^{2} distribution of the statistic s^{2} will give us the confidence interval for σ^{2}:^{[49]}
where t_{k,p} and χ2 k,p are the pth quantiles of the t- and χ^{2}-distributions respectively. These confidence intervals are of the confidence level1 − α, meaning that the true values μ and σ^{2} fall outside of these intervals with probability (or significance level) α. In practice people usually take α = 5%, resulting in the 95% confidence intervals.
Approximate formulas can be derived from the asymptotic distributions of $\textstyle {\hat {\mu }}$ and s^{2}:
The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_{α/2} do not depend on n. In particular, the most popular value of α = 5%, results in |z_{0.025}| = 1.96.
Normality testsEdit
Normality tests assess the likelihood that the given data set {x_{1}, ..., x_{n}} comes from a normal distribution. Typically the null hypothesisH_{0} is that the observations are distributed normally with unspecified mean μ and variance σ^{2}, versus the alternative H_{a} that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:
Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
Q–Q plot, also known as normal probability plot or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it's a plot of point of the form (Φ^{−1}(p_{k}), x_{(k)}), where plotting points p_{k} are equal to p_{k} = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
P–P plot – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_{(k)}), p_{k}), where $\textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}$. For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Shapiro–Wilk test: This is based on the fact that the line in the Q–Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
Tests based on the empirical distribution function:
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
Either the mean, or the variance, or neither, may be considered a fixed quantity.
When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
Both univariate and multivariate cases need to be considered.
This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:
The factor ${\frac {ay+bz}{a+b}}$ has the form of a weighted average of y and z.
${\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that ${\frac {ab}{a+b}}$ is one-half the harmonic mean of a and b.
Vector formEdit
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size $k\times k$, then
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_{i}x_{j}=x_{j}x_{i}$, only the sum $a_{ij}+a_{ji}$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form $\mathbf {x} '\mathbf {A} \mathbf {y} =\mathbf {y} '\mathbf {A} \mathbf {x} .$
where ${\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}$
With known varianceEdit
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows $x\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ with known variance σ^{2}, the conjugate prior distribution is also normally distributed.
This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ^{2}. Then if $x\sim {\mathcal {N}}(\mu ,1/\tau )$ and $\mu \sim {\mathcal {N}}(\mu _{0},1/\tau _{0}),$ we proceed as follows.
First, the likelihood function is (using the formula above for the sum of differences from the mean):
In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean ${\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}$ and precision $n\tau +\tau _{0}$, i.e.
That is, to combine n data points with total precision of nτ (or equivalently, total variance of n/σ^{2}) and mean of values ${\bar {x}}$, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)
The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows $x\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ^{2} is as follows: