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I can not see why moment generating functions are useful. What are the applications? I know they give an alternate representation of a probability functions and I guess they can be easier to work with in some cases, but are there any other applications?

Amzoti
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    to add to your list: establishing probability bounds here and there ... – Math-fun Jan 20 '15 at 13:19
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    Well, calculating expected values of powers of a variable (which indirectly helps in calculating variances). There's also Chernoff bounds. Oh, and given two random variables $X$ and $Y$ such that $M_X = M_Y$, then $X = Y$. – Misguided Jan 20 '15 at 14:33

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