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March 18th, 2018, 01:46 PM  #1 
Member Joined: Feb 2018 From: Canada Posts: 42 Thanks: 2  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: $$ P_{X,Y}(x,y) = \begin{cases}0 &\text{if X=0 and Y is finite} \\ P(X=0)=\dfrac{\dbinom{20}{0}\dbinom{80}{300}}{\dbinom{100}{30}} &\text{if X=0 and Y is infinite} \\P_X(x)P_{Y/X}(y/x) = \dfrac{\dbinom{20}{x}\dbinom{80}{30x}}{\dbinom{100}{30}} \dbinom{301}{41} 0.8^{304} 0.2^4 &\text{if }X=1,2,\dots ,20 \\ 0 &\text{ otherwise} \end{cases} $$ Last edited by skipjack; March 20th, 2018 at 05:22 PM. 

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binomial, distribution, hypergeometric, negative, pmf 
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