Worked Example 1

Suppose \({{X}{1}},{{X}{2}},...,{{X}_{n}}\) is a random sample of size \(n\) taken from a population with mean \(\mu\) and variance \({{\sigma }^{2}}\). If \(\bar{X}\) is the sample mean, show that \(\mathrm{E}\left( {\bar{X}} \right)=\mu \) and \(\mathrm{Var}\left( {\bar{X}} \right)=\frac{{{\sigma }^{2}}}{n}\).