Hypergeometric Distribution

Caution

Hypergeometric distribution has no relation to the geometric distribution.

Hypergeometric Experiment

From a population of size $N$ containing $k$ successes, a sample of size $n$ is retrieved without replacement. Number of successes in the sample is observed and denoted by $X$.

The hypergeometric distribution describes the probability of obtaining $x$ successes in a hypergeometric experiment. Denoted by $h(x;N,n,k)$.

$$P(x) = \frac{^kC_x \times\; ^{N-k}C_{n-x}}{^NC_n}$$

Here:

• $^kC_x$ - Number of ways to choose $x$ successes from $k$ successes.
• $^{N-k}C_{n-x}$ - Number of ways to choose $n-x$ failures from $N-k$ failures.
• $^NC_n$ - Total number of ways to choose $n$ items from $N$ items.

Mean

$$\mu_x = \frac{nk}{N}$$

Variance

$$V_x = \frac{nk(N-k)(N-n)}{N^2 (N - 1)}$$

Cumulative Hypergeometric Distribution

Refers to the probability that the hypergeometric random variable is greater than a lower limit or lesser than an upper limit.

Multivariate Hypergeometric Distribution

Suppose a population of size $N$, having $k$ different types of items. Each type has $N_1, N_2, \ldots, N_k$ items.

$$\sum_{i=1}^{k} N_i = N$$

Multivariate Hypergeometric Distribution describes the probability of obtaining $x_1, x_2, \ldots, x_k$ items of each type from the above population. Denoted by $h(\mathbf{x};N,n,\mathbf{N})$.

$$P(x_1, x_2, \dots, x_k) = \frac{\binom{N_1}{x_1} \cdot \binom{N_2}{x_2} \cdot \ldots \cdot \binom{N_k}{x_k}}{\binom{N}{n}}$$