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Let X, Y be two independent random continuous variables, that is:

$$ f(x, y) = f_X(x)\cdot f_Y(y) $$

where $f_X$ and $f_Y$ are their respective marginal density functions.

I've read a lot of times that the expected value of the product of these variables is the product of their expected values, that is

$$ E(XY)=E(X)E(Y) $$

I would like to find a demonstration of this fact that didn't involve the Law of Total Expectation. For example:

$$ \begin{align} E(XY) &=\iint_D x\cdot y \cdot f(x, y) \;dy\;dx\\ &=\iint_D x\cdot y \cdot f_X(x) \cdot f_Y(y) \;dy\;dx\\ &= \dots \\ &= \int_D x \cdot f_X(x)\;dx \int_D y \cdot f_Y(y)\;dy\\ &= E(X)E(Y) \end{align} $$

What properties of double integrals am I missing to fill the gap?

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    Observe that $\int k~g(x) \operatorname d x = k~\int g(x)\operatorname d x$ where the integral's bound variable, $x$, does not occur in $k$. Then you can 'extract' the inner integral. – 2017-01-28
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    Thank you @GrahamKemp, your comment let me fill in the blank. It seems so obvious now... I guess taking the Probability course too much time after the Calculus one has its disadvantages... – 2017-01-29

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