Sampling Distributions – Advanced

Property 1: If x is a random variable with N(μ, σ²) distribution and samples of size n are chosen, then the sample mean has a normal distribution N(μ, σ²/n).

Proof: We start by looking at the moment-generating function of x̄. Since the x_i in a sample are independent, by Properties 2 and 3 of General Properties of Distributions

But since the x_i are taken from a random sample, they all have the same probability density function, namely that of the normal distribution N(μ, σ). Thus

and so

But the right side is the moment generating function for N(µ, ), and so the result follows by Theorem 1 of General Properties of Distributions.

References

Soch, J. (2020) Proof: Moment-generating function of the normal distribution. The book of statistical proofs
https://statproofbook.github.io/P/norm-mgf.html

Hoel, P. G. (1962) Introduction to mathematical statistics. Wiley