Gaussian distribution, also known as normal distribution, also known as normal distribution . For a random variable X, its probability density function is shown in the figure. The distribution is called Gaussian distribution or normal distribution, denoted as N(μ, σ2), where are the parameters of the distribution, which are the expectation and variance of the Gaussian distribution, respectively. When there is a certain value, p(x) is also determined, especially when μ=0, σ2=1, the distribution of X is a standard normal distribution . The μ normal distribution was first obtained by De Moivre in 1730 when he was looking for the asymptotic formula of the binomial distribution; Post-Laplace was also introduced when he studied the limit theorem in 1812; Gauss was in 1809 when he studied It was also derived from the error theory. The function image of the Gaussian distribution is a bell-shaped curve located above the x-axis, which is called the Gaussian distribution curve, or Gaussian curve for short .
In 1809, Carl Friedrich Gauss (1777-1855) published his famous book "The Theory of Motion of Celestial Bodies Around the Sun". At the end of the book, he writes a section on the problem of "data combination", which actually involves the determination of this error distribution.
He did the same as Laplace. But as he went down, he came up with two innovative ideas. One is that he does not adopt a Bayesian inference method. The measurement error is formed by many factors, each of which has little effect. According to the central limit theorem , it is inevitable that its distribution approximates the normal distribution. In fact, as early as around 1780, Laplace generalized Demoffer's result and obtained a more general form of the central limit theorem . Unfortunately, he failed to apply this result to the problem of determining the error distribution. The second innovative idea of Gauss is: he reverses the problem, first admits that the arithmetic mean is the estimate that should be taken , and then finds the error density function . This is the normal distribution. A probability distribution. The normal distribution is the distribution of a continuous random variable with two parameters μ and σ2 . The first parameter μ is the mean of the random variable that follows the normal distribution, and the second parameter σ2 is the variance of this random variable, so the normal distribution Denoted as N(μ,σ2). The probability law of a random variable that follows a normal distribution is that the probability of taking a value near μ is high, and the probability of taking a value farther away from μ is smaller; dispersion. The characteristic of the density function of the normal distribution is that it is symmetrical about μ, reaches a maximum value 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, σ2=1, it is called the standard normal distribution, denoted as N(0, 1). When a μ-dimensional random vector has a similar probability law, the random vector is said to follow a multi-dimensional normal distribution. The multivariate normal distribution has nice properties, for example,The marginal distribution of the multivariate normal distribution is still a normal distribution, and the random vector obtained by any linear transformation is still a multidimensional normal distribution, especially its linear combination is a univariate normal distribution.
The normal distribution was first obtained by A. De Moivre in the asymptotic formula for the binomial distribution. CF Gauss derived it from another angle when studying measurement error . PS Laplace and Gauss studied its properties.
This work of Gauss had a great influence on later generations. He gave the normal distribution the name of "Gaussian distribution" at the same time. The reason why later generations attributed the invention of the least squares method to him is also due to this work. Gauss was a great mathematician with important contributions too numerous to list. But today's German 10-mark banknotes with Gaussian faces also have a normally distributed density curve printed on them. This conveys the idea that, of all the scientific contributions of Gauss, this one has had the greatest impact on human civilization.