Questions about the sampling distribution of the sample mean

(a) Let $X_1,X_2,\cdots,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$.

Show that

$E[\bar{X}]=\mu$ and $V[\bar{X}]=\frac{\sigma^2}{n}$

What is the distribution of $\bar{X}$?

(b) If the distribution of $X_1,X_2,\cdots,X_n$ were not Normal, under what conditions can the distribution of $\bar{X}$ be approximated by a Normal distribution?

• Do you have any insights into the problem by yourself? As is, you're just asking for an answer. Apr 5 '13 at 2:18
• Don't trust rules like "$n>30$"... Jun 4 '13 at 8:30

(a) $E[\bar{X}]$ = $E[\frac{X_1 + X_2 + X_3+\dots+X_n}{n}$]
= $\frac{1}{n}E[X_1+X_2+X_3+\dots + X_n]$
=$\frac{1}{n}(\mu_1 + \mu_2 +\mu_3+\dots +\mu_n) = \frac{1}{n}(n\mu) = \mu$