Sampling Distribution (4) Distribution of sample mean and sample variance

Distribution of sample mean and sample variance in normal population

With total X, regardless of distribution, as long as the mathematical expectation e (x) and Variance D (x) exist

Set its mathematical expectation (mean) E (x) =μ, Variance D (x) =σ2

X1,x2,x3,x4,x5 ..... Xn is a sample from the total X, and the sample mean and sample variance are:

Proposition one has a total of X, its mathematical expectation e (x) =μ, Variance D (x) =σ2,x1,x2 ... Xn is a sample from the overall x, the

　　　　　　　　　　　　　　

So

(1) The expected value of the sample mean is equal to the expectation of total X, the average size of the sample mean is equal to the average size of the total x value.

(2) The variance of the sample mean is equal to one of the N of the total x variance. It is indicated that when the sample capacity n increases, the variance of the sample mean is much smaller than the variance of the population, and the concentration of the sample mean on the mathematical expectation μ is much higher than the concentration degree of the whole. When n is large, the sample mean is dense near μ.

(3) The mean value of the sample variance is the variance of the population.

Distribution of sample mean and sample variance for a normal population distribution

Sampling Distribution (4) Distribution of sample mean and sample variance