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Let X∼Pois(19).

a) Find the mgf of X−EX

b) Find E(X-EX)^3

I'm not really sure how to go about finding the MGF of X-EX. I know the MGF of poisson distribution, but not sure how to use that to find X-EX.

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Use the definition of MGF. For $t\in\mathbb{R}$ such that the MGF is well-defined, $$ M_{X-\mathbb{E}[X]}(t) = \mathbb{E}[ e^{t(X-\mathbb{E}[X])}] = \mathbb{E}[ e^{tX}e^{-t\mathbb{E}[X]}] = e^{-t\mathbb{E}[X]}\mathbb{E}[ e^{tX}] = e^{-t\mathbb{E}[X]}M_X(t) $$ where we used the fact that $e^{-t\mathbb{E}[X]}$ is just a scalar, so that we can "pull it out" of the expectation.

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