Question

5. Suppose X. N(μ, σ2), what is the distribution of the sample mean Σ ? Comment on the behavior of the distribution for increasing n. Furthermore, is the distribution of the sample mean consistent with the predictions of the central limit theorem?

Answer 1

Proof of the distribution of Sample mean:

Given X N(H,o) ID's The MGF of XisM.'(t)=e' 2 ,i=1, 2,3 n and E (Xi)-μ and Var(Xi)-d, 1,2,3.. n The MGFof Xis 12 which is MGF of normal distributon with mean μ and variance Mean of the sampling distribution of X is Variance of the sampling di stribution of X is trr 2 i-1 2 i-1 Therefore,\ X ~ N/ μ

i.e. The sampling distribution of sample mean follows normal distribution with mean mu and variance sigma^2/n

Central Limit Theorem

The central limit theorem states that:

Given a population with a finite mean ? and a finite non-zero variance ?², the sampling distribution of the mean approaches a normal distribution with a mean of ? and a variance of ?²/N as N, the sample size, increases.

Proof of Consistent:

As per sufficient conditi on for consistent, i) lim E(x)-lim μ-μ ii) lim Var(x)lm0 Therefore, x is Consistent esim ator of μ