Let ${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{x}}_{\mathbf{n}}$be independent random variables, each with density function $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, expected value$\mathit{\mu }$ , and variance${\mathit{\sigma }}_{\mathbf{2}}$ . Define the sample meanby.$\stackrel{\mathbf{-}}{\mathbf{x}}\mathbf{=}\sum _{i=1}^n{x}_i$Showthat$\mathit{E}\mathbf{\left(}\stackrel{\mathbf{-}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{\mu }$,and .$\mathit{v}\mathit{a}\mathit{r}\mathbf{\left(}\stackrel{\mathbf{-}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{\sigma }}^{\mathbf{2}}}{\mathbf{n}}$ (See Problems $\mathbf{5}\mathbf{.9}\mathbf{,}\mathbf{5}\mathbf{.13}$and$\mathbf{6}\mathbf{.15}$.)

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