The question is
Let $X_1,…,X_n$ be drawn iid from $Beta(0.1,0.5)$. Let $\bar{X} = \frac{1}{n}\sum^n_{i=1} X_i$.
a) Derive $\mathbb{E}(\bar{X})$ and $\mathbb{V}(\bar{X})$
I know how to get the expectation and variance of a random variable from a given distribution. But I'm not how to get the expectation and variance of $\bar{X}%$. Not sure where to being actually.
Best Answer
I got the same answer as you. However, be aware of the following:
$ \mathbb{V}(X + Y) = \mathbb{V}(X) + \mathbb{V}(Y) + 2\text{COV}(X,Y) $
For a long summation of variances, like the one you did, it becomes:
$ \mathbb{V}(\sum^n_{i=1}X_i)=\sum^n_{i=1}\sum^n_{j=1}\text{COV}(X_i,X_j) $
This is the "more complicated" part that whuber mentioned :-). Make sure you understand what happens to the covariances and why. Let me know if you need help there.