Hypergeometric

The hypergeomtric distribution is given as,

\[ P(X=g; n, G, N) = \frac{{G \choose G}{N-G \choose n-g}}{N \choose n} \]

Expected Value : $\(\mathbb E[X] = np; \quad p = \frac{G}{N}\)$

Variance : $\( \text{Var}[X] = \underbrace{npq}_{\text{Binom. Var}} \frac{N-n}{N-1} \)$

Sums of Dependent Bernoulli Trials

The hypergeometric is the sum of \(n\) non-identical \(\text{Bernoulli}(G/N)\) trials,

\[ X = \sum I_k \]

\[\begin{split} I_k \sim \text{Bernoulli}(G_k/N_k)\\ \end{split}\]

The dependency comes from the fact that the hypergeometric process samples without replacement. On the other hand, its cousin is the binomial process that samples with replacement hence the independency.