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I've only seen the MGF used when it takes 0 as it parameter and then derived for the nth moment.

What does the MGF tell us when the parameter is other than 0?

Veta
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    For an idea of what it can be used for have a look at this posts: http://stats.stackexchange.com/questions/72479/general-sum-of-gamma-distributions/137318#137318 http://stats.stackexchange.com/questions/183313/bound-for-weighted-sum-of-poisson-random-variables/183355#183355 – kjetil b halvorsen Dec 26 '15 at 18:33

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