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1. Consider n independent ?ips of a coin having probability p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the ?ips are H H T H T H H T, then there are a total of ?ve changeovers. If p = 1/2, what is the probability there are k changeovers?

2. Let X be a Poisson random variable with parameter λ. Show that P{X = i} increases monotonically and then decreases monotonically as i increases, reaching its maxi- mum when i is the largest integer not exceeding λ.

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