I have two random variables $X$ and $Y$. The random variable have non-negative support meaning that their PDF is defined only for non negative values. I know the PDF's of $X$ and $Y$. I want to find the expectation of $XY^j$ (where $j$ is a real number) given that $XY^j
Answer
To be clear, you are looking for $\mathsf E(XY^j\mid XY^j I get: $$\begin{align}\mathsf E(XY^j\mid XY^j
\\[1ex] ~=~& \bbox[gainsboro]{\dfrac{\int\limits_0^\infty~x~\mathsf E( Y^j~\mathbf 1_{Y<(w/x)^{1/j}}\mid X=x)~f_X(x)~\mathrm d x }{\int\limits_0^\infty~\mathsf P(Y<(w/x)^{1/j}\mid X=x)~f_X(x)~\mathrm d x }}
\\[1ex] ~=~& \dfrac{\int\limits_0^\infty~x~f_X(x)\int\limits_0^{(w/x)^{1/j}} y^j~f_{Y\mid X}(y\mid x)~\mathrm d~y~\mathrm d x }{\int\limits_0^\infty~f_X(x)\int\limits_0^{(w/x)^{1/j}} f_{Y\mid X}(y\mid x)~\mathrm d~y~\mathrm d x }
\end{align}$$
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