What we can say is that: \(\displaystyle{E}{\left({X}{Y}\right)}={E}{\left({X}{E}{\left({Y}{\mid}{X}\right)}\right)}\)

But we really do need to know what the joint distribution is to say more (or at least a marginal and conditional distribution).

\(\displaystyle{E}{\left({X}{Y}\right)}=\int\int_{{{\mathbb{{{R}}}}^{{2}}}}{x}{y}{{f}_{{{X},{Y}}}{\left({x},{y}\right)}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}=\sum_{{x}}\sum_{{y}}{x}{y}{P}{\left({X}={x},{Y}={y}\right)}\)

\(\displaystyle{E}{\left({\left({X}{E}{\left({Y}{\mid}{X}\right)}\right)}=\int_{{{\mathbb{{{R}}}}}}{x}{{f}_{{X}}{\left({x}\right)}}\int_{{{\mathbb{{{R}}}}}}{y}{{f}_{{{Y}{\mid}{X}={x}}}{\left({y}\right)}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}=\sum_{{y}}{P}{\left({X}={x}\right)}\sum_{{y}}{y}{P}{\left({Y}={y}{\mid}{X}={x}\right)}\right.}\)

And so forth.