Home page for accesible maths 5 Models for discrete random variables 5.7 Negative Binomial random variables (not examinable)5.9 Summary

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5.8 Hypergeometric random variables (not examinable)

A lake contains an unknown number $N$ of fish. A sample of $n$ fish is drawn from the lake and each of these fish is marked and replaced. A week later a second sample of $k$ fish is drawn from the lake and is found to include exactly $m$ marked fish. We wish to know how many fish $N$ there are in the lake. For example, if we marked $n=20$ fish, then came back and saw that $m=8$ out of $k=32$ fish we looked at were marked, it suggests that $8/32=1/4$ of the fish in the lake were marked. If 20 is a a quarter of the fish in the lake, it suggests there were $N=80$ fish in the lake, but can we model this properly to work out how certain we are of our guess for $N$ ? To summarise:

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${N}$ fish in the lake,
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${n}$ are marked and replaced,
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${k}$ drawn the second time of which ${R}$ are observed to have been previously marked.

	$\displaystyle{\rm P}(R=r)$	$\displaystyle=$	$\displaystyle\frac{\mbox{ways of choosing ${r}$ marked and ${k-r}$ unmarked}}{% \mbox{ways of choosing ${k}$}}$
		$\displaystyle=$	$\displaystyle\frac{\binom{n}{r}\binom{N-n}{k-r}}{\binom{N}{k}}.$

Properties are more tricky mathematically than the Binomial or Poisson.

Example 5.25.

Assume that ${n=3}$ fish are marked, and that ${k=5}$ fish are drawn in the second catch, and there are ${N=7}$ fish in total Write down the pmf of ${R}$ .

Solution.

For $r=1,2,3$

	$\displaystyle p_{R}(r)$	$\displaystyle=$	$\displaystyle\frac{\binom{3}{r}\binom{7-3}{5-r}}{\binom{7}{5}}$
		$\displaystyle=$	$\displaystyle\frac{1}{21}{\binom{3}{r}\binom{4}{5-r}}$