We have $n$ different types of coupons in the box. Probability of drawing coupon of $j$th type from the box does not change and is equal to $p_j$. Find the expected value and variance of number of types in randomly chosen set of $k$ coupons.
I've managed to find the expected value, but I cannot calculate the variance. This is what I've got so far.
Let $X_i = \begin{cases} 0, & \text{if}\ \text{the $i$th coupon was not found in $k$ selections} \\ 1, & \text{otherwise} \end{cases}$
We have $X = \sum_{i=1}^{n} X_i$ and $E[X_i] = 1 \times Pr(X_i = 1) + 0 = 1 - (1 - p_i)^k$. $E[X] = E[\sum_{i=1}^{n}X_i] = \sum_{i=1}^nE[X_i] = n - \sum_{i=1}^{n}(1 - p_i)^k$.
Now let's calculate variance. $\text{Var}\left[X\right] = E[X^2] - (E[X])^2$. Since I've managed to calculate $E[X]$ the only problem is to obtain $E[X^2]$.
So $E[X^2] = E[(\sum_{i=1}^{n}X_i)^2] = \sum_{i=1}^{n}E[X_i^2] + 2\sum_{1 \leq i < j \leq n}E[X_iX_j]$.
The first summand is quite straightforward, but the second one certainly not. Could you suggest me any solution?