Distribution

Hypergeometric Distribution

We use the Hypergeometric Distribution to answer the following question:

• Given $N$ total objects, where $r$ of them are considered successes, and in $n$ trials, what is the probability you draw $k$ successes without replacement, i.e. $P(X=k)$?

Hypergeometric Distribution (Definition)

We say $X\sim HGeo(N,r,n)$, where:

1. $X$ is the number of observed successes.
2. $n$ objects are randomly selected from $N$ total objects without replacement.
3. There are only objects of two types: successes or failures. There are $r$ successes.

p.m.f. of $X$: $P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{r}{k}\right)\left(\genfrac{}{}{0ex}{}{N-r}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$

Expectation and Variance

• $E(X)=\frac{nr}{N}$
• $Var(X)=\frac{nr}{N}(1-\frac{r}{N})\frac{N-n}{N-1}$

If it was with replacement, we can convert $HGeo(N,r,n)$ to $Bin(n,r/N)$, the Binomial Distribution

Related

• Negative Binomial Distribution