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March 18th, 2018, 12:46 PM   #1
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The P.M.F. of Hypergeometric and Negative Binomial Distribution.

A forest contains 100 deer. 20 of them have a red tag and 80 of them are untagged. A researcher samples 30 random deer without replacement. Let X be the number of tagged deer in the sample. From the sample of 30 deer, she will keep picking deer with replacement until she picks the fourth tagged deer. Let Y be the number of selections she makes until she gets her fourth tagged deer. Find the joint pmf of X and Y.

This is my solution; can someone take a look for me? Thank you.
Let X be the number of tagged deer in the sample of 30 random deer without replacement.
$$X \sim \text{Hypergeometric(}N=100, n=30, r=20).$$
Let Y/X be the number of selections from the sample of 30 deer she makes until she gets her fourth tagged deer.
$$Y/X \sim \text{Negative Binomial(}r=4, p=0.2).$$
The joint P.M.F. of X and Y is as follows:
$$
P_{X,Y}(x,y) = \begin{cases}0 &\text{if X=0 and Y is finite} \\ P(X=0)=\dfrac{\dbinom{20}{0}\dbinom{80}{30-0}}{\dbinom{100}{30}} &\text{if X=0 and Y is infinite} \\P_X(x)P_{Y/X}(y/x) = \dfrac{\dbinom{20}{x}\dbinom{80}{30-x}}{\dbinom{100}{30}} \dbinom{30-1}{4-1} 0.8^{30-4} 0.2^4 &\text{if }X=1,2,\dots ,20 \\ 0 &\text{ otherwise} \end{cases}
$$

Last edited by skipjack; March 20th, 2018 at 04:22 PM.
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