Define hypergeometric distribution. Find its mean and variance.
Answers
hypergeometric distribution is a discrete probability distribution that describes the probability of {\displaystyle k} k successes (random draws for which the object drawn has a specified feature) in {\displaystyle n} n draws, without replacement, from a finite population of size {\displaystyle N} N that contains exactly {\displaystyle K} K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of {\displaystyle k} k successes in {\displaystyle n} n draws with replacement.
I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution.
If a box contains N balls, a of them are black and N−a are white, and n number of balls are drawn at random without replacement , then the probability of getting x black balls (and obviously n−x white balls) is given by the following p.m.f.
The p.m.f is
f(x)=(aCx)⋅(N−aCn−x)NCn
The mean is given by:
μ=E(x)=np=na/N
and, variance
σ2=E(x2)+E(x)2=na(N−a)(N−n)N2(N2−1)=npq[N−nN−1]
where
q=1−p=(N−a)/N
I want the step by step procedure to derive the mean and variance. Thank you.