M.g.f for Poisson distribution

Trillian

New Member
Hello,
I am having trouble understanding the maths between E($$e^{tx}$$) where I get a summation from 0 to $$\infty$$ and the m.g.f for Poisson distribution. The notes I have are to break out $$e^{-\lambda}/e^{-\lambda e^t}$$
(Now, don't get me wrong - I DO get how to make this last expression easier on the eye - It's just that breaking this out from the sum makes the sum go towards 1 as x approaches $$\infty$$)
I can't seem to find what is supposed to be left in the summation nor why that would behave so nicely when the whole expression looks so strange (to me). I have been to (http://mathworld.wolfram.com/PoissonDistribution.html) and other resources, but my problem is not finding the mgf or using it but understanding how to get to it myself from the probability mass function.

BGM

TS Contributor
$$X \sim \mathrm{Poisson}(\lambda)$$

$$E[e^{tX}] = \sum_{x=0}^{+\infty} e^{tx} e^{-\lambda} \frac {\lambda^x} {x!} = e^{-\lambda} \sum_{x=0}^{+\infty} \frac {(e^t\lambda)^x} {x!} = e^{-\lambda} \exp\{e^t\lambda\} = \exp\{\lambda(e^t - 1)\}$$

It is not difficult; You just need to know the series for exponential function
(or just the kernel of the Poisson distribution)

Trillian

New Member
going back to my calculus book again with new leads - Thank you!

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