Hypergeometric distribution

Introduction

We have already studied that in binomial experiment the trials are independent and the probability of success remains constant from trails to trials. So the binomial distribution can be applied to the sampling with replacement. But if in an experiment, the probability of success doesnt remain the same from one trial to another, then the binomial distribution can not be applied. In this case, an approptiate distribution is hypergeometric distribution.

An experiment, in which the evente are stochastically dependent, although random, where the probability of success does not remain the same from trial to trial, is called hypergeometric experiment. In other words, an experiment of sampling without replacement from a finite population is known as hhypergeometric experiment. So, in such experiment we draw a random sample of size n from a population of size N without replacement. Then we are interested to find the probability of selecting x items having a certain characteristics from the total number of items with the same characteristics in the population. The random variable X = Numbers of items having a certain cheracteristic i.e. number of successes, in the random sample of size n is called hypergeometric random variable and its probability distribution is called hypergeometric distribution.

Defination :

A random variable X is said to have a hypergeometric distribution with three parameters N, M and n if it assumes only non-negative values and its probability mass function (p.m.f) is

$$P(X=x) \ = \ P[(x; N, M, n)] \ = \frac{\binom{M}{x} \ \binom{N-M}{n-x}}{\binom{N}{n}} \ \ \ ; \ x \ = \ 0, \ 1, \ 2, \ ... \ min \ (n,M)$$

where N is the population size (finite), M is the total number of items with a certain characteristic in the population and n is the sample size where, M≤ N and n≤ N.

Derivation of hypergeometric distribution

Suppose that we have a population of N objects of which M objects have a certain characteristics, called M success and N-M objects do not have the characteristics, called N-M failures. Let X be the number of successes found in the sample. $$Since, \ x \ successes \ can \ be\ drawn \ from \ M \ successes \ in \binom{M}{x} \ ways \ and \ (n-x) \ failures \ can \ be \ drawn \ from \ the \ remaining \ N-M \ failures \ in \binom{N-M}{n-x} \ ways, \ the \ total \ number \ of \ favourable \ cases \ to \ the \ selection \ of \ x \ successes \ and \ n-x \ failures \ is \binom{M}{x} \ \binom{N-M}{n-x}.$$

$$Again, \ the \ n \ objects \ from \ the \ N \ objects \ can \ be \ drawn \ in \binom{N}{n} \ ways, \ which \ is \ the \ exhaustive \ cases.$$

Then the probability of selecting x successes from the M successes and (n-x) failures from the N-M failures is given by

$$P(X= x) \ = \ p(x) \ = \ p(x; N, M, n) \ = \frac{ \binom{M}{x} \binom{N-M}{n-x} }{\binom{N}{n}} \ \ \ \ ; x = 0, 1, 2, ... min (n,M)$$

This is the probability mass function of hypergeometric distribution with three parameters N, M and n.

It can easily be shown that

$$\sum_{x=0}^n p(x; N, M, n) \ = \sum_{x=0}^n \frac{ \binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}}$$

$$= \frac{ \binom{N}{N}}{\binom{N}{n}}$$

= 1

Alternative from of hypergometric distribution

An alternative expeaaionfor the hypergeometric distribution is

$$ p(n_1) \ = \frac{ \binom{N_1}{n_1} \binom{N_2}{n_2}}{\binom{N}{n}} \ \ \ \ ; \ n_1 \ = 0, \ 1, \ 2, \ ... \ n$$

Where the population size, N = N₁ + N₂; N₁ = Number of successes, N₂ = Number offailures in the population and sample size, n = n₁ + n₂; n₁ and n₂ are the number of successes and failures respectively in the sample.

Geleralized form of hypergeometric distribution

Let a population of N objects consists of N₁ objectsof type 1, N₂, objects of types 2, ... N_k objects of type k where N₁ + N₂+ ... + N_k = N. If a random sample of size n is drawn without replacement from the N_i objects of that type so that n₁ + n₂ + ... n_k = n. Then the probability of selecting the sample with the specified composition is

$$p(n_1, n_2, ..., n_k) \ = \frac{ \binom{N_1}{n_1} \ \binom{N_2}{n_2} \ ... \binom{N_k}{n_k}}{ \binom{N}{n}}$$

where, n_i≤ N_i

The distribution having the above probability mass function is called generalized or multi variate hypergeometric distribution.

moments of hypergeometric distribution

Let the random variable X has a hypergeometric distribution with probability mass function

$$p(x; N, M, n) \ = \frac{ \binom{M}{x} \ \binom{N-M}{n-x}}{ \binom{N}{n}} \ \ \ \ ; x = 0, \ 1, \ 2, \ ..., \ min(n,m)$$

Now,

$$E(x) \ = \sum_{x=0}^n x \ p(x; N, M, n)$$

$$= \sum_{x=0}^n \ x \frac{ \binom{M}{x} \binom{N-M}{n-m}}{\binom{N}{n}}$$

$$= \frac{1}{\binom{N}{n}} \ \sum_{x=0}^n \ x \frac{M!}{x! (M-x)!} \ \binom{N-M}{n-x}$$

$$= \frac{M}{\binom{N}{n}} \ \sum_{x=1}^n \ x \frac{(M-1)!}{(x-1)! (M-x)!} \ \binom{N-M}{n-x}$$

$$= \ \frac{M}{\binom{N}{n}} \ \sum_{x=1}^n \binom{M-1}{x-1} \ \binom{N-M}{n-x}$$

$$= \ \frac{M}{\binom{N}{n}} \ \binom{N-1}{n-1}$$

$$\therefore \ E(x) \ = \ n \frac{M}{N}$$

$$Hence, \ mean \ of \ hypergeometric \ distribution \ is n\frac{M}{N}$$

$$If, \ p \ = \frac{M}{N} , \ then \ E(x) = np$$

Again,

E(x²) = E[ x (x-1) + x]

= E[X(X-1)] + E(X)

$$E[ x (x-1)] \ = \sum_{x=0}^n \ x \ (x-1) \ p(x; N, M, n)$$

$$= \ \sum_{x=0}^n \ x (x-1) \frac{ \binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}}$$

$$= \ \frac{1}{\binom{N}{n}} \ \sum_{x=0}^n \ x(x-1) \ \frac{M!}{x!(M-x)!} \ \binom{N-M}{n-x}$$

$$ = \ \frac{M(M-1)}{\binom{N}{n}} \ \sum_{x=2}^n \ \frac{(M-2)!}{(x-2)! (M-x)!} \ \binom{N-M}{n-x}$$

$$ = \frac{M(M-1)}{\binom{N}{n}} \ \sum_{x=2}^n \ \binom{M-2}{x-2} \ \binom{N-M}{n-x}$$

$$= \ M(M-1) \frac{n(n-1)}{N(N-1)}$$

$$\therefore \ E(x²) = \ \frac{M(M-1) n (n-1)}{N(N-1)} \ \frac{nM}{N}$$

Hence,

V(x) = E(x²) - [E(x)]²

$$= \ \frac{nM(N-M)(N-n)}{N^2(N-1)}$$

Hence the variance of the hypergeometric distribution is

$$ V(x) = \ npq ( \frac{N-n}{N-1})$$

$$where, \ p = \frac{M}{N} \ and \ q \ = 1-p \ = \frac{N-M}{N}$$

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.