1. (2pt) Recall that the PMF of Poisson(\) is P(X = k) = e¯\ת², k = 0, 1, 2, ……. . (a) Let X ~ Poisson (A) and consider the sequence p(0), p(1), p(2),... with p(k) = P(X = k). Show that p(k) first increases and then decreases¹. At what k does the change happen? (b) Let X ~ Bin(n, p) and consider the sequence p(0), p(1),...,p(n) with p(k) = P(X = k). Show that p(k) first increases and then decreases. At what k does the change happen? p(k) Hints. Consider the ratio (+1). Using integer parts (floor/ceiling functions) can be helpful for simplifying your answer.

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