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I roll a fair, 6-sided dice twice. Let $X$ and $Y$ be the outcomes of my first and second rolls. I subsequently find the moment generating function of $S = X + Y$ by finding $E(e^{t(X+Y)})$.

Why does $E(e^{t(X + Y)}) = E(e^{tX})E(e^{tY})$? I can see how $E(e^{t(X + Y)}) = E(e^{tX}e^{tY})$, but why can we just separate this expression into the product of two expected values?

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    Because $X$ and $Y$ are assuned independent. If not, that separation would NOT be possible! It is a general theorem that for independent random variables $X$ and $Y$ then $\mathrm{E} (XY) = \mathrm{E(X) \mathrm{E}(Y)$.2012-12-09

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We can do it because of independence.

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    Although I think the identity in _some_ cases holds when $X$ and $Y$ are not independent.2012-12-09