Expectation of the mean of the sum of random variables

Question

If $X_i$'s are independent and identified random variables, each with mean $\mu$ and variance $\sigma^2$. Let's say $S_m = \frac{1}{m} \sum_{i=1}^m X_i,~~ m = 1,2,\ldots,M.$ What are the values of $\mathbb {E}[S_m]$ and $Var(S_m)$?

Answer 1

Recall that $E(X+Y) = E(X) + E(Y)$ hence we have:
$$E(S_m) = \frac{m*\mu}{m}$$ $$E(S_m) = \mu $$ Recall that $Var(X+Y) = Var(X) + Var(Y)$. Also recall that $Var(cX) - c^2 Var(x)$ hence we have:
$$Var(\frac{1}{m} \sum_{i=1}^m X_i,~~ m = 1,2,\ldots,M) = m \sigma^2$$ $$Var(S_m) = \frac{m \sigma^2}{m^2}$$ $$Var(S_m) = \frac{\sigma^2}{m}$$

I hope that helps.

Bob