Hypergeometric Distribution
- $(\Omega, \mathcal{F}, P)$,
- $N \in \mathbb{N}$,
- \(K \in \{0, 1, \ldots, N\}\),
- \(n \in \{1, \ldots, N\}\),
- \(X: \Omega \rightarrow \{0, \ldots, N\}\),
p.d.f. of hypergeometric function is defined by
\[\begin{eqnarray} k \in \{0, 1, \ldots, N\}, \ \mathrm{Hyper}(k; N, K, n) & := & \frac{ \left( \begin{array}{c} K \\ k \end{array} \right) \left( \begin{array}{c} N - K \\ n - k \end{array} \right) }{ \left( \begin{array}{c} N \\ n \end{array} \right) } \end{eqnarray}\]- $k$,
- is the number of observed succsess
- $n$,
- is the number of trials
c.d.f.
\(\begin{eqnarray} F_{\mathrm{Hyper}}(k; N, K, n) & := & \sum_{i=0}^{k} \mathrm{Hyper}(i; N, K, n) \end{eqnarray}\)
\[\begin{eqnarray} \sum_{i=k + 1}^{N} \mathrm{Hyper}(i; N, K, n) & = & 1 - F_{\mathrm{Hyper}}(k; N, K, n) \end{eqnarray}\]Example
- $N$,
- the number of total groups
- $K$,
- the number of people in YES condition
- $n$,
- the number of people in treatment group
| Condition | treatment group | control group | total |
|---|---|---|---|
| YES | $k$ | $K − k$ | $K$ |
| NO | $n − k$ | $N + k − n − K$ | $N − K$ |
| total | $n$ | $N − n$ | $N$ |
Suppose that $k$ is given by a value of some r.v. $X$. Then p.d.f of $X$ follows $\mathrm{Hyper}(k; N, K, n)$. Indeed, there are ${K \choose k}$ patterns to become YES state, ${N - K \choose n - k }$ patterns to become NO state and ${ N \choose n}$ total patterns to be in treatment group from $N$.
Proposition
\[\forall k \in \{0, 1, \ldots, N\}, \ \max(0, n + K - N) \le k \le \min(K, n) \Rightarrow \mathrm{Hyper}(k; N, K, n) \ge 0 .\]proof
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