Show the expectation Let $X_1,\ldots,X_n$ be i.i.d random variables with mean $\mu$ and variance $\sigma^2$ Let $S_n=\frac{(X_1+\cdots+X_n)}{n}$ and $V_n=\frac{1}{n-1} \sum_{i=1}^n (X_i-S_n)^2$. Show that $E(S_n) =\mu$ and $E(V_n)=\sigma^2$
Would the variance be tackled something like this:
$\operatorname{Var}(X)= \operatorname{E}(X-\operatorname{E}(X))^2$ then $\operatorname{E}\left(\frac{\sum_{i=1}^n (X_i-S_n)^2)}{n-1}\right) = \operatorname{E}\left(\frac{\sum_{i=1}^n (X_i-S_n)^2}{n-1}\right)-\operatorname{E}\left(\frac{\sum_{i=1}^n (X_i-S_n)^2)}{n-1}\right)^2$
 A: It is easy to see that $E(S_n) = \mu$. To see $E(V_n)=\sigma^2$,
$$ \begin{align}
E(X_i - S_n)^2 &= E(X_i^2 + S_n^2 -2X_i S_n) \\
&= E(X_i^2) + E(S_n^2)-2E(X_iS_n) \quad \cdots (1)
\end{align}
$$ 
Since $E(X_i^2) = \mu^2 +\sigma^2$ and $E(S_n^2) = \mu^2 + n^{-1}\sigma^2$, we have 
$$
\begin{align}
(1) &= 2\mu^2 + (1+\frac{1}{n})\sigma^2 - \frac{2}{n}E\left(\sum_{k=1}^{n}X_iX_k \right) \\
&= 2\mu^2 + (1+\frac{1}{n})\sigma^2 - \frac{2}{n}\left[\sum_{k=1, k\neq i}^n \big(E(X_i)E(X_k)\big) + E(X_i^2)\right] \\
&=2\mu^2 + (1+\frac{1}{n})\sigma^2-\frac{2}{n}\left((n-1)\mu^2 + \mu^2 +\sigma^2 \right) \\
&= (1-\frac{1}{n})\sigma^2. 
\end{align}
$$
Thus, 
$$
\begin{align}
E(V_n) &= \frac{1}{n-1} \sum_{i=1}^{n} E(X_i-S_n)^2 \\
&= \frac{1}{n-1} n (1-\frac{1}{n})\sigma^2 \\
&=\sigma^2.
\end{align}$$

Since $X_i, 1\leq i \leq n$ are uncorrelated, $var(\sum_{i=1}^{n} X_i)=\sum_{i=1}^n var(X_i).$ So,
$$
\begin{align}
var(S_n)&=\frac{1}{n^2} var\left(\sum_{i=1}^{n} X_i\right) \\
&=\frac{1}{n^2} \sum_{i=1}^{n} var(X_i) \\
&= \frac{\sigma^2}{n}.
\end{align}
$$
Thus, $E(S_n^2) = E(S_n)^2 + var(S_n) = \mu^2 + n^{-1} \sigma^2.$
A: You have $$n^2E(S_n^2) = E(\sum X_i^2 + 2\sum X_iX_j) = \sum EX_i^2 + 2\sum EX_iEX_j = \sum(\sigma^2+\mu^2) +2\sum \mu^2 = n\sigma^2 + n\mu^2 + n(n-1)\mu^2 = n\sigma^2 + n^2\mu^2.$$
Then $ES_n^2 = \mu^2 + \frac{1}{n}\sigma^2.$
Note that $(X_i-\mu)^2 = (X_i-S_n)^2 + (S_n-\mu)^2 + 2(X_i-S_n)(S_n-\mu)$. Then $$\sum_{i=1}^n E(X_i-\mu)^2 = \sum_{i=1}^n E(X_i-S_n)^2 + nE(S_n-\mu)^2$$ 
$$n\sigma^2 = E\left(\sum_{i=1}^n(X_i-S_n)^2\right) + n(ES_n^2-\mu^2)$$
$$n\sigma^2 = (n-1)EV_n + \sigma^2$$
$$EV_n = \sigma^2.$$
