Finding Variance of the Hypergeometric distribution

Question

I've been trying to find the variance of the Hypergeometric distribution, but have had issues calculating $\ E [X^2]$. Can anyone help?

Answer 1

For a Hypergeometric Distribution, with population $N$, population favoured items $K$, and sample size $n$.

Let $X_i$ be the indicator that the $i$-th item in the sample is favoured, for $i\in\{1, .., n\}$

$$X=\sum_{i=1}^n X_i$$

We note that $\mathsf P(X_i{=}1)= K/N$ and $\mathsf P(X_i{=}1, X_j{=}1) = K(K-1)/N(N-1)$ (for all $i\neq j)$ and because $X_\star$ are indicator random variables:

$$\mathsf E(X_i) = \frac KN$$

$$\mathsf E(X_iX_{j:j\neq i}) = \frac {K(K-1)}{N(N-1)}$$

Then $$\mathsf E(X^2) = \mathsf E\left(\left(\sum_{i=1}^n X_i\right)\left(\sum_{j=1}^n X_j\right)\right) =\quad\lower{2ex}\ldots$$