(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?

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

(a) $E[\bar{X}]$ = $E[\frac{X_1 + X_2 + X_3+\dots+X_n}{n}$]

= $\frac{1}{n}E[X_1+X_2+X_3+\dots + X_n]$

=$\frac{1}{n}(\mu_1 + \mu_2 +\mu_3+\dots +\mu_n) = \frac{1}{n}(n\mu) = \mu$

maybe you could work on the the next part and see if you get stuck. Here is the wikipedia about the central limit theorem that should help with the third question in part (a) and part (b)


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