Question: 2. Let X be a Poisson random variable with parameter > 0, probability function 1e-A x! and moment generating function M(t) = (-1) (a)

2. Let X be a Poisson random variable with parameter > 0,

2. Let X be a Poisson random variable with parameter > 0, probability function 1e-A x! and moment generating function M(t) = (-1) (a) Noting the Taylor Series approximation f(x) = e" =1+v+ x = 0, 1, 2,... 23 + 21 3! show that f(r) is a valid probability function. (b) Evaluate E(X) from first principles. (Hint: note a! 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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