Hypergeometric distribution

In theory of probability and statistics, the hypergeometric distribution is a discreet probability distribution related to random sampling and no replacement. Suppose you have a population of N{displaystyle N} of which, K{displaystyle K} belong to the category A{displaystyle A} and N− − K{displaystyle N-K} belong to the category B{displaystyle B}. Hypergeometric distribution measures the likelihood of obtaining x{displaystyle x} (0≤ ≤ x≤ ≤ K{displaystyle 0leq xleq K}) elements of the category A{displaystyle A} in a sample without replacement n{displaystyle n} elements of the original population.

Definition

Probability Function

A discreet random variable X{displaystyle X} has a hypergeometric distribution with parameters N=0,1,...... {displaystyle N=0,1,dots }, K=0,1,...... ,N{displaystyle K=0,1,dotsN} and n=0,1,...... ,N{displaystyle n=0,1,dotsN} and write X♥ ♥ HG (N,K,n){displaystyle Xsim operatorname {HG} (N,K,n)} if your probability function is

P [chuckles]X=x]=(Kx)(N− − Kn− − x)(Nn),{displaystyle operatorname {P} [X=x]={frac {{K choose x}{N-K choose n-x}}{N choose n}}},}

for values x{displaystyle x} between max{0,n− − N+K!{displaystyle max{0,n-N+K}} and min{K,N− − K!{displaystyle min{K,N-K}}where N{displaystyle N} is the size of population, n{displaystyle n} is the size of the sample extracted, K{displaystyle K} is the number of elements in the original population belonging to the desired category and x{displaystyle x} is the number of elements in the sample that belong to that category.

The notation

(ba)=b!a!(b− − a)!{displaystyle {b choose a}={frac {b!}{a!

refers to the binomial coefficient, i.e. the number of possible combinations when selecting a{displaystyle a} elements of a total b{displaystyle b}.

Recursive formula

Yeah. X♥ ♥ HG (N,K,n){displaystyle Xsim operatorname {HG} (N,K,n)} Then it can be proved that

P [chuckles]X=x+1]=(K− − x)(n− − x)(x+1)(N− − K− − n+x− − 1)P [chuckles]X=x]{displaystyle {begin{aligned}operatorname {P} [X=x+1] stranger={frac {(K-x)}{(n-x)}{(x+1)(N-K-n+x-1)}}}}}{operatorname {P} [X=x]end{aligned}}}}}}}}

Properties

Yeah. X♥ ♥ HG (N,K,n){displaystyle Xsim operatorname {HG} (N,K,n)} then. X{displaystyle X} fulfills some properties:

The expected value of the random variable X{displaystyle X} That's it.

E [chuckles]X]=nKN{displaystyle operatorname {E} [X]={frac {nK}{N}}

and its variance is given by

Var [chuckles]X]=nKN(N− − KN)(N− − nN− − 1){displaystyle operatorname {Var} [X]={frac {nK}{n}{bigg} ({frac {N-K}{n}}{bigg)}{bigg (}{bigg {n}{N-n}{N-1}{bigg}}}}}}{bigg

The hypergeometric distribution is applicable to sampling without replacement and the binomial to sampling with replacement. In situations where the expected number of repetitions in the sample is presumably low, the first can be approximated by the second. This is so when N is large and the relative size of the drawn sample, n/N, is small.

Related Distributions

• If a random variable X♥ ♥ HG (N,K,1){displaystyle Xsim operatorname {HG} (N,K,1)} then. X♥ ♥ Bernoulli (KN){displaystyle Xsim operatorname {Bernoulli} left({frac {K}{N}{N}right)}.
• Yeah. X♥ ♥ HG (N,K,n){displaystyle Xsim operatorname {HG} (N,K,n)} then. X♥ ♥ Binomial (n,p){displaystyle Xsim operatorname {Binomial} (n,p)} When N→ → ∞ ∞ {displaystyle Nto infty } and K→ → ∞ ∞ {displaystyle Kto infty} in such a way that K/N→ → p{displaystyle K/Nto p}.