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Fall 2011 Strategic Practice 7: Section 1 (Joint, Conditional, and Marginal Distributions) - Question 5
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A chicken lays n eggs. Each egg independently does or doesn't hatch, with probability p of hatching. For each egg that hatches, the chick does or doesn't survive (independently of the other eggs), with probability s of survival. Let N ~ Bin(n, p) be the number of eggs which hatch, X be the number of chicks which survive, and Y be the number of chicks which hatch but don't survive (so X + Y = N). Find the marginal PMF of X, and the joint PMF of X and Y. Are they independent?

Solution:
Marginally we have X ~ Bin(n, ps). (X, Y, Z) has a Multinomial(n, (ps, p(1−s), (1−p)). Note: Here X and Y are not independent, unlike in the chicken-egg problem from class (where N was Poisson). This follows immediately from thinking about an extreme case: if X = n, then clearly Y = 0.