Alternative Derivation of Poisson MGF

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Here is the alternative derivation of the Poisson MGF discussed in class:

First, recall that the exponential function satisfies:

e^\theta = \sum_{x=0}^{\infty} \frac{\theta^x}{x!}

So if X is a Poisson random variable, its

M_X(t) \sum_{x=0}^{\infty} e^{-\lambda} \frac{ (\lambda e^t)^x}{x!} = e^{-\lambda} \sum_{x=0}^{\infty} \frac{ (\lambda e^t)^x }{x!}

Using the result for exponential functions this equals

e^{-\lambda} e^{e^t \lambda} = e^{e^t \lambda - \lambda}.

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