Question: 2. Let X be a Poisson random variable with parameter > > 0, probability function f (20) = ac! x = 0, 1, 2, ..

2. Let X be a Poisson random variable with parameter > > 0, probability function f (20) = ac! x = 0, 1, 2, .. . and moment generating function M(t) = ex(et-1). (a) Noting the Taylor Series approximation 82 2 2 ev = 1 + v+ + .. 2! 3! acc ! x=0 show that f(x) is a valid probability function. (b) Evaluate E(X) from first principles. Hint: note x! in the denominator of f (x), and follow the same approach as for the binomial random variable.] (c) Confirm the mean using the moment generating function of X

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