(a) Show that X + Y ~ Bin(n + m, p), using a story proof.

(b) Show that X - Y is not Binomial.

(c) Find P(X = k|X + Y = j). How does this relate to the elk problem from HW 1?

Solution: (a) Interpret X as the number of successes in n independent Bernoulli trials and Y as the number of successes in m more independent Bernoulli trials, where each trial has probability p of success. Then X + Y is the number of successes in the n + m trials, so X + Y ~ Bin(n + m, p). (b) A Binomial can't be negative, but X - Y is negative with positive probability. (c) Using the definition of conditional probability to find P(X = k|X + Y = j) we get (nCk)(nC[j-k])/([n+m]Cj). This is exactly the same distribution as in the elk problem (it is called the Hypergeometric distribution). To see why, imagine that there are n male elk and m female elk, each of which is tagged with the word "success" with probability p (independently). Suppose we then want to know how many of the male elk are tagged, given that a total of j elk have been tagged. For this, p is no longer relevant, and we can "capture" the male elk and count how many are tagged, analogously to the original elk problem.

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