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

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

But since the xi 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, $\sigma/\!\sqrt{n}$), and so the result follows by Theorem 1 of General Properties of Distributions.