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If we have two random Variables that are independent and identically distributed, their sum's moment generating function is simply their product. But does the trick work if they are not identically distributed? For example, if one is Normally distributed, and the other is Exponentially distributed, can we still multiply the moment generating functions?

I can't see any reason it won't work, but have not been able to find any examples on the Internet, which is why I wanted to confirm this.

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Answered in comments:

The trick works – Dilip Sarwate

More details:

Wikipedia on moment generating functions gives a slightly more general result in an obvious place to look (what were you searching for?). Also see math.SE: Moment generating function of $X+Y$ using convolution of $X$ and $Y$. – Glen_b

MGFs and characteristic functions are conceptually the same. Concerning the latter, see and visit 1 for worked examples. – whuber

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