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\begin{equation*}
\begin{aligned}
E\{X Y\} &= \int_{-\infty|}^\infty \int_{-\infty}^\infty x y p(x) p(y)\, dx\, dy \\
  &= \int_{-\infty}^\infty \left( \int_{-\infty}^\infty x p(x)\, dx \right) y p(y)\, dy
  = \int_{-\infty}^\infty \mu_X y p(y) dy = \mu_X \mu_Y.
\end{aligned}
\end{equation*}
\end{document}