= Solution
Interpret existence of the <expectations> as finite <expected values>, equivalently $\mathbb E|X|<\infty$ and $\mathbb E|Y|<\infty$. First justify the summations rather than assuming that a signed double series can be rearranged. <Independence> and <Tonelli theorem> give
$$
\sum_{x,y}|xy|p_{X,Y}(x,y)=\sum_{x,y}|x|p_X(x)|y|p_Y(y)=\mathbb E|X|\,\mathbb E|Y|<\infty.
$$
Consequently $XY$ is an <integrable random variable>, and <Fubini theorem> permits summing the signed terms in either order:
$$
\mathbb E[XY]=\sum_x\sum_yxy\,p_X(x)p_Y(y)=\left(\sum_xxp_X(x)\right)\left(\sum_yyp_Y(y)\right)=\boxed{\mathbb E[X]\mathbb E[Y].}
$$
If extended infinite <expectations> are allowed instead, the phrase “exist” needs additional care: expressions such as $0\cdot\infty$ do not define an <expectation>. The finite-<expectation> convention avoids that ambiguity.
Back to article page
