A Poisson random variable has a PMF of the form

(a) Find the characteristic function,ϕX( ω ) .
(b) Find the first three nonzero terms in the Taylor series expansion of ln[ϕX(ω)].
(c) Use the results of part (b) to find the mean, variance, and skewness of the Poisson random variable.

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Pr(k) = uĪ exp (-a), k = 0, 1, 2,