Properties of the sample mean

Summary

• Given an IID random sample \(x_1,x_2,...,x_n\) where \(E(x_i)=μ\) , \(Var(x_i)=σ^2\) for \(i=1…n\) , the sample mean

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}{ ‍ }x_i\]

• has the following properties:

\[E(\bar{x})=μ\]

\[Var(\bar{x})= \frac{σ^2}{n}\]

• Given an IID random sample of size \(n\) from \(N(μ,σ^2)\) we have

\[\bar{x} \sim N(μ,σ^2/n)\]