Hypergeometric Distribution

A sample of size n is chosen at random from an urn containing N balls of which m are white. This is called a draw without replacement.

The random variable we are going to study is X the number of white balls selected:

\begin{displaymath}P\{X=i \}=\frac{\binom{m}{i}\binom{N-m}{n-i}}{\binom{N}{n}}\end{displaymath}

This only takes on positive values for:

\begin{displaymath}n-(N-m)\leq i \leq min(n,m)\end{displaymath}

Example:
Tagging animals, N unknown, we conduct a tagging experiment by catching and marking m animals and recapture i of them that are tagged, out of the recapture sample of size n. We find it by maximising the probability of P(X=i)=Pi(N).

To see which N maximises this we notice that :

\begin{displaymath}\frac{P_i(N)}{P_i(N-1)}=\frac{(N-m)(N-n)}{N(N-m-n+i)}\end{displaymath}

This ratio is greater than 1 if and only if
\begin{displaymath}(N-m)(N-n)\geq N(N-m-n+i) \mbox{ equivalent to } N\leq \frac{mn}{i} \end{displaymath}

This increases and then decreases and is max at [mn/i]. This is what is called the maximum likelihood estimate of N.

Example:
There are 50 tagged deer in the forest, mark them and release them, a subsequent catch n=40, of which 4 are found to have marks then $\hat{N}=500$.

Remark: If we supposed that probability of finding tagged animal is binomial, we get the same conclusion.