Normal distribution (Gaussian distribution)

Normal distribution （Normal distribution）, Also known as “ Normal distribution ”, Also known as Gaussian distribution （Gaussian distribution）, First by Timothy （Abraham de Moivre） In the asymptotic formula of binomial distribution, we get .C.F. Gauss derived it from another angle when he studied the measurement error .P.S. Laplace and Gauss studied its properties . It's one in math 、 Probability distribution is very important in physics and engineering , Has a significant impact on many aspects of Statistics .
The normal curve is bell shaped , Both ends are low , Middle high , Left right symmetry because of its bell shaped curve , So people often call it bell curve .
If random variable X Obey a mathematical expectation of μ、 The variance of σ2 Is a normal distribution , Write it down as N(μ,σ2). The probability density function is the expected value of the normal distribution μ It determines its position , The standard deviation σ It determines the magnitude of the distribution . When μ = 0,σ = 1 The normal distribution of time is the standard normal distribution .

The properties of important quantities of Gaussian distribution

The density function is symmetric about the mean value, and the mean value is its mode （statistical mode） And the median （median） Under function curve 68.268949% The area of , Within a range of standard deviation around the mean 95.449974% The area of , Two standard deviations around the mean 2σ Within the scope of 99.730020% The area of , Three standard deviations around the mean 3σ The scope of the , Among them the first 3-5 Article is called 68-95-99.7 The laws of . For example ：