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  • Sample Mean and Sample Variance

Sample Mean

• Let $X_1,\dots ,X_n$ be a Random Sample from a distribution with $\mathrm{E}[X_i]=\mu$ and $\mathrm{Var}(X_i)={\sigma }^2$
• Define the sample mean as $\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$. Then $\mathrm{E}[\overline{X}]=\mu$ and $\mathrm{Var}(\overline{X})=\frac{{\sigma }^2}{n}$
• $E[X_i]=\mu ,Var(X_i)={\sigma }^2(i=1,\dots ,n)$
• Define the sample mean
• $\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$
• Expectation
  • $E[\overline{X}]=E\left[\frac{1}{n}\sum _{i=1}^n{X}_i\right]=\frac{1}{n}\sum _{i=1}^{n}E[X_i]=\frac{1}{n}\sum _{n=1}^n\mu =\mu$
  • So the sample mean is an unbiased estimator of the population mean.

Proof

• $\mathrm{E}[\overline{X}]=\mathrm{E}\left[\frac{1}{n}\sum _{i=1}^n{X}_i\right]$
• By linearity
• $=\frac{1}{n}\sum _{i=1}^n\mathrm{E}[X_i]$
• Sub in
• $=\frac{1}{n}\sum _{i=1}^n\mu$
• $=\frac{1}{n}n\mu =\mu$