# 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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