Problem
Let $X_1,...,X_n$ be a random sample of a distribution $X$ with expected value $\mu$ and variance $\sigma^2$. Show that $\overline{X}^2$ is not an unbiased estimator of $\mu^2$. Is it asintotically unbiased? Is it consistent?
I got stuck at the very beginning of this exercise. First, to prove that $\overline{X}^2$ is not unbiased, I should show that $E[\overline{X}^2] \neq \mu^2$.
By definition, $$\overline{X}=\dfrac{X_1+...+X_n}{n},$$ so $$\overline{X}^2=\dfrac{1}{n^2}(\sum_{i=1}^n {X_i}^2+2\sum_{i<j}X_iX_j)$$
Now, I want to calculate the expected value of the expression of the right. I am not sure how to this, any help would be greatly appreciated.
Hint: $$\mathbb E[\overline{X}^2] = \text{Var}(\overline{X}) + \mathbb E[\overline{X}]^2$$ Do you know the mean and variance of $\overline{X}$?