Question: A random variable X has the Poisson distribution p(x; ) = ex/x! for x = 0, 1, 2,... . Show that the moment-generating function of

A random variable X has the Poisson distribution p(x; μ) = e−μμx/x! for x = 0, 1, 2,... . Show that the moment-generating function of X is MX(t) = e

μ(et−1).

Using MX(t), find the mean and variance of the Poisson distribution.

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