Distribution of $\bar{X}$ of n Bernoulli's

Question

I am trying to derive the distribution of $\bar{X}_n$ where $X_1, X_2,...,X_n$ are iid $\sim \mathrm{Bern}(p)$. I used two approaches but I am debating myself and questioning which one would be correct (if any).

Method 1: Using MGF I used the moment generating function and ended up with $\bar{X}_n \sim \mathrm{Bern}(p^n)$

Method 2: I used the CLT and ended up with $\bar{X}_n \sim N(p, \sqrt{pq}/n$) for n being large.

I am not sure which one is correct (if any).

Can someone tell me if I am doing this right or not?

Thank you, I appreciate your help

Answer 1

Neither.

• For the first one, you are claiming that the average only take binary value.

• For the second one, what if $n$ is small.

Guide:

• Recall that sum of IID Bernoulli follows a binomial distribution $Bin(n, p)$.
• Average is simply dividing the sum by $n$.