Worked Example 1

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