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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

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 2 {\displaystyle \sigma ^{2}} . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

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

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

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

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

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):

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 \varphi } (phi).[8] The alternative form of the Greek letter phi, {\displaystyle \varphi } , is also used quite often.

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 2 {\displaystyle \sigma ^{2}} . The precision is normally defined as the reciprocal of the variance, 1 / 2 {\displaystyle 1/\sigma ^{2}} .[10] The formula for the distribution then becomes

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.

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

About 68% of values drawn from a normal distribution are within one standard deviation tag_hash_119 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.

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 normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance.[16][17] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.[18][19]

The normal distribution is a subclass of the 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.

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

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

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.

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.

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[38] (Matlab code). In the following sections we look at some special cases.

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.

This result is known as Cramr's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramr'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.[31]

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.

By Cochran's theorem, for normal distributions the sample mean ^ {\displaystyle \textstyle {\hat {\mu }}} and the sample variance s2 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 ^ {\displaystyle \textstyle {\hat {\mu }}} and s can be employed to construct the so-called t-statistic:

The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles ztag_hash_128/2 do not depend on n. In particular, the most popular value of tag_hash_130 = 5%, results in |z0.025| = 1.96.

Normality tests assess the likelihood that the given data set {x1, ..., xn} comes from a normal distribution. Typically the null hypothesis H0 is that the observations are distributed normally with unspecified mean tag_hash_135 and variance tag_hash_1362, versus the alternative Ha that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

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

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