. Suppose that N has a Poisson distribution with parameter λ. (a) Show that the probabilities in (3.2) sum to 1. [Hint: This uses the expansion of ex that you learned in calculus.] (b) Use the same...

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Suppose that N has a Poisson distribution with parameter λ.

(a) Show that the probabilities in (3.2) sum to 1. [Hint: This uses the expansion

of ex that you learned in calculus.]

(b) Use the same expansion (by taking the first and second deriviatives) to

show EN = λ and VarN = λ.

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Verify that if U has the uniform distribution on (a, b), then EU =

(a + b)/2 and Var U = (b − a)2/12.