Question: 1. Let X Bin(n1, p), Y Bin(n2, p) be independent and let Z = X + Y. (a) Find the conditional PMF of

1. Let X ∼ Bin(n1, p), Y ∼ Bin(n2, p) be independent and let Z = X + Y.

(a) Find the conditional PMF of Z given that X = k.

(b) Use the result in part (a), and the Law of Total Probability for marginal PMFs as was done in Example 5.3-1, to provide an analytical proof of Example 5.3-2, namely that Z ∼ Bin(n1+n2, p). (Hint.

You will need the combinatorial identity

/n1+n2 k

0 1n1 =

i=0

/n1 i

0/ n2 k−i 0

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