HYPER-GEOMETRIC DISTRIBUTION

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Hyper-geometric Distribution

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◇Overview

The Hypergeometric Distribution models the number of successes $X$ in a sample of size $n$ drawn without replacement from a finite population of size $N$ that contains exactly $K$ successes (and $N - K$ failures). Unlike the Binomial distribution, trials are dependent because each draw alters the composition of the remaining population.

◇Key Definitions

• Probability Mass Function (PMF):
  $P(X = x) = \frac{\binom{K}{x} \binom{N - K}{n - x}}{\binom{N}{n}}, \quad \max(0, n - (N - K)) \le x \le \min(n, K)$
• Mean: $E[X] = n \cdot \frac{K}{N} = np$ (where $p = \frac{K}{N}$).
• Variance: $\text{Var}(X) = n \cdot \frac{K}{N} \left(1 - \frac{K}{N}\right) \left(\frac{N - n}{N - 1}\right)$, where $\frac{N - n}{N - 1}$ is the Finite Population Correction (FPC) factor.

◇Worked Example

Problem: A shipment of $N = 10$ routers contains $K = 3$ defective units and $7$ working units. If an inspector tests a random sample of $n = 4$ routers without replacement, what is the probability that exactly $x = 2$ tested routers are defective?

Solution:
$P(X = 2) = \frac{\binom{3}{2} \binom{7}{2}}{\binom{10}{4}} = \frac{3 \times 21}{210} = \frac{63}{210} = 0.30$

◇References & Navigation

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• Related Notes: ◇Discrete Probability Distribution, ◇Hypothesis Testing