In fact, the mean is u and the variance is t 2.
So: ∫ e [-(x-u) 2/2 (t 2)] dx = (√ 2π) t (*)?
The integral region is from negative infinity to positive infinity, and the following integral is also in this region.
(1) average?
Derivation of u on both sides of formula (*):
∫{e^[-(x-u)^2/2(t^2)]*[2(u-x)/2(t^2)]dx=0？
Round the constant, and then multiply both sides by 1/(√2π)t to get:
∫[ 1/(√2π)t]*e^[-(x-u)^2/2(t^2)]*(u-x)dx=0？
Disassemble (u-x) and move items:
∫x*[ 1/(√2π)t]*e^[-(x-u)^2/2(t^2)]dx=u*∫[ 1/(√2π)t]*e^[-(x-u)^2/2(t^2)]dx？
What is that?
∫x*f(x)dx=u* 1=u？
In this way, the definition of mean value is given and it is proved that the mean value is u.
(2) variance?
The process is about the same as the average, so I'll just write a little.
Derivation of t on both sides of formula (*):
∫[(x-u)^2/t^3]*e^[-(x-u)^2/2(t^2)]dx=√2π？
Mobile project:
∫[(x-u)^2]*[ 1/(√2π)t]*e^[-(x-u)^2/2(t^2)]dx=t^2？
What is that?
∫(x-u)^2*f(x)dx=t^2？
The definition of variance has just been worked out and the conclusion has been proved.
Extended data:
If the random variable X obeys the normal distribution with a mathematical expectation of μ and a variance of σ 2, it is recorded as N(μ, σ 2). The expected value μ of probability density function with normal distribution determines its position, and its standard deviation σ determines its distribution amplitude. When μ = 0 and σ = 1, the normal distribution is standard normal distribution.
In statistical description, variance is used to calculate the difference between each variable (observed value) and the population mean. In order to avoid the phenomenon that the average sum deviation is zero and the average square sum deviation is affected by the sample size, the average deviation of the average square sum is used to describe the variation degree of variables.
Because the image of a normal population is not necessarily symmetrical about Y, for any normal population, its value is less than the probability of X, as long as it can be used to find the probability of a normal population in a certain interval.
For the convenience of description and application, normal variables are often converted into data. Convert a general normal distribution into a standard normal distribution.
For continuous random variable X, if its definition domain is (a, b) and the probability density function is f(x), the formula for calculating the variance of continuous random variable X is: d (x) = (x-μ) 2 f (x) dx?
Variance describes the dispersion degree between the value of random variable and its mathematical expectation. (The greater the standard deviation and variance, the greater the dispersion)
If the values of x are concentrated, the variance D(X) is small, and if the values of x are scattered, the variance D(X) is large.
Therefore, D(X) is a quantity to describe the dispersion degree of X and a scale to measure the dispersion degree of X..
Baidu encyclopedia-variance
Baidu Encyclopedia-Normal Distribution