The first parameter μ is the mean of a random variable that obeys normal distribution, and the second parameter σ2 is the variance of this random variable, so the normal distribution is recorded as N(μ, σ2). The probability law of random variables subject to normal distribution is that the probability of taking values near μ is high, and the probability of taking values far from μ is low; The smaller σ is, the more concentrated the distribution is around μ, and the larger σ is, the more dispersed the distribution is. The density function of normal distribution is characterized by: with respect to μ symmetry, it reaches the maximum at μ, takes a value of 0 at positive (negative) infinity, and has an inflection point at μ σ. Its shape is high in the middle and low on both sides, and the image is a bell curve above the X axis. When μ = 0 and σ 2 = 1, it is called standard normal distribution, and it is recorded as n (0, 1). When a μ-dimensional random vector has similar probability laws, it is said that this random vector follows a multidimensional normal distribution. Multivariate normal distribution has good properties, such as the edge distribution of multivariate normal distribution is still normal distribution, and the random vector obtained by arbitrary linear transformation is still multidimensional normal distribution, especially its linear combination is unary normal distribution.

The normal distribution was first obtained by A. de moivre in the asymptotic formula of binomial distribution. C.F. Gauss deduced it from another angle when studying the measurement error. Laplace and Gauss studied its properties.

The probability distribution of many random variables in production and scientific experiments can be approximately described by normal distribution. For example, in the case of unchanged production conditions, the strength, compressive strength, caliber, length and other indicators of the product; Body length, weight and other indicators of the same organism; Weight of the same seed; Measuring the error of the same object; Deviation of the impact point in a certain direction; Annual precipitation in a certain area; And the velocity component of ideal gas molecules, and so on. Generally speaking, if a quantity is the result of many tiny independent random factors, then it can be considered to have a normal distribution (see the central limit theorem). Theoretically, normal distribution has many good properties, and many probability distributions can be approximated by it. There are also some commonly used probability distributions directly derived from it, such as lognormal distribution, T distribution, F distribution and so on.