Sample mean

Definition:

• Some what expected value of a sample
• $\bar{X}=\sum _{i=1}^n\frac{X_i}{n}$
• Proof:
  • Let $X_1,...,X_n$ be independent and identically distributed random variables having distribution function $F$ and expected value µ.
  • Such a sequence of random variables is said to constitute a sample from the distribution $F$.
  • Or different combination of observations of the same distribution is also a sample
  • Then the quantity $\bar{X}=\sum _{i=1}^n\frac{X_i}{n}$
• $E[\bar{X}]=\mu$, mean of sample mean is the population mean

Variance of sample mean: $Var(\bar{X})=\frac{{\sigma }^2}{n}$