Standard Normal Distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:$f(x) = \frac e^$
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. 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). For other names, see Naming.

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

Definitions

Standard normal distribution

The simplest case of a normal distribution is known as the ''standard normal distribution'' or ''unit normal distribution''. This is a special case when $\mu=0$ and $\sigma =1$, and it is described by this probability density function (or density):
:$\varphi(z) = \frac$
The variable $z$ has a mean of 0 and a variance and standard deviation of 1. The density $\varphi(z)$ has its peak $1/\sqrt$ 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
:$\varphi(z) = \frac$
which has a variance of 1/2, and Stephen Stigler once defined the standard normal as
:$\varphi(z) = e^$
which has a simple functional form and a variance of $\sigma^2 = 1/(2\pi)$

General normal distribution

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

:$f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac \sigma \right)$

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 $X$.

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 $\phi$ (phi). 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 $\mathcal(\mu,\sigma^2)$. Thus when a random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, one may write

:$X \sim \mathcal(\mu,\sigma^2).$

Alternative parameterizations

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, $1/\sigma^2$. The formula for the distribution then becomes

:$f(x) = \sqrt e^.$

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

: $f(x) = \frac e^.$

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 $\textstyle\theta_1=\frac$ and $\textstyle\theta_2=\frac$, and natural statistics ''x'' and ''x''². The dual expectation parameters for normal distribution are and .

Cumulative distribution functions

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

:$\Phi(x) = \frac 1 \int_^x e^ \, dt$

The related error function $\operatorname(x)$ gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range x, x/math>. That is:

:$\operatorname(x) = \frac 2 \int_0^x e^ \, dt$

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

: \Phi(x) = \frac \left + \operatorname\left( \frac x \right) \right/math>

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

:
F(x) = \Phi\left(\frac \sigma \right) = \frac \left + \operatorname\left(\frac\right)\right

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

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:

:\Phi(x)=\frac + \frac\cdot e^ \left + \frac + \frac + \cdots + \frac + \cdots\right/math>

where $!!$ 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:

$\Phi(x) \approx \frac+\frac\sum_^\frac$

Standard deviation and coverage

About 68% of values drawn from a normal distribution are within one standard deviation ''σ'' away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. 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
:$F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname \left(\frac\right).$
To 12 significant figures, the values for $n=1,2,\ldots , 6$ are:

For large $n$, one can use the approximation $1 - p \approx \frac$.

Quantile function

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:
:$\Phi^(p) = \sqrt2\operatorname^(2p - 1), \quad p\in(0,1).$
For a normal random variable with mean $\mu$ and variance $\sigma^2$, the quantile function is
:$F^(p) = \mu + \sigma\Phi^(p) = \mu + \sigma\sqrt 2 \operatorname^(2p - 1), \quad p\in(0,1).$
The quantile $\Phi^(p)$ 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 $z_$ 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. Note that the following table shows $\sqrt 2 \operatorname^(p)=\Phi^\left(\frac\right)$, not $\Phi^(p)$ as defined above.

For small $p$, the quantile function has the useful asymptotic expansion
$\Phi^(p)=-\sqrt+\mathcal(1).$

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.Geary RC(1936) The distribution of the "Student's" ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184

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 derivatives

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.
* 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 $f^\prime(x)=-\frac f(x).$
* 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 $x=\mu+\sigma.$
* Its density is log-concave.
* Its density is infinitely differentiable, indeed supersmooth of order 2.

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 $\varphi^(x)=(x^2-1)\varphi(x)$
* More generally, its th derivative is $\varphi^(x) = (-1)^n\operatorname_n(x)\varphi(x),$ where $\operatorname_n(x)$ is the th (probabilist) Hermite polynomial.
* 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.

Moments

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:
:
\operatorname\left X-\mu)^p\right=
\begin
0 & \textp\text \\
\sigma^p (p-1)!! & \textp\text
\end

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 $p,$

:\begin
\operatorname\left confluent_hypergeometric_functions_$_1F_1$_and_$U.$

:\begin
___\operatorname\left ^p\right&=_\sigma^p\cdot_(-i\sqrt_2)^p_U\left(-\frac,_\frac,_-\frac_\left(_\frac_\mu_\sigma_\right)^2_\right),_\\
___\operatorname\left Hermite_polynomials#"Negative_variance", generalized_Hermite_polynomials.

The_expectation_of_$X$_conditioned_on_the_event_that_$X$_lies_in_an_interval_ ,b/math>_is_given_by
:\operatorname\left _\mid_a=_\mu_-_\sigma^2\frac_
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_function_

The_ Fourier_transform_of_a_normal_density_$f$_with_mean_$\mu$_and_standard_deviation_$\sigma$_is

:$\hat_f(t)_=_\int_^\infty_f(x)e^_\,_dx_=_e^_e^$

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_$e^$,_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_$t$._The_relation_between_both_is:
:$\varphi_X(t)_=_\hat_f(-t)$

__Moment_and_cumulant_generating_functions_

The_moment_generating_function_of_a_real_random_variable_$X$_is_the_expected_value_of_$e^$,_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

:M(t)_=_\operatorname ^=_\hat_f(it)_=_e^_e^

The_ cumulant_generating_function_is_the_logarithm_of_the_moment_generating_function,_namely

:$g(t)_=_\ln_M(t)_=_\mu_t_+_\tfrac_12_\sigma^2_t^2$

Since_this_is_a_quadratic_polynomial_in_$t$,_only_the_first_two_cumulant_
In_probability_theory_and_statistics,_the_cumulants__of_a_probability_distribution_are_a_set_of_quantities_that_provide_an_alternative_to_the_''_moments''_of_the_distribution.__Any_two_probability_distributions_whose_moments_are_identical_will_have_...