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

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

• For the second one, what if $$n$$ is small.
• Recall that sum of IID Bernoulli follows a binomial distribution $$Bin(n, p)$$.
• Average is simply dividing the sum by $$n$$.
• Suppose $n=2$, the values that you can get are $0, 0.5, 1$. It is possible to get value $0.5$, something that is not modelled by the Bernoulli distribution. – Siong Thye Goh Dec 4 '18 at 1:12