If $X$ is a random variable with pdf $f(x)$, what is $\mathbb E [Xg(X)]$

Question

I have two random variables $X$ and $Y$ and want to determine $\mathbb E[XY]$. The random variable $X$ has a continuous pdf on $\mathbb R$, $f$. The relation between $X$ and $Y$ is given by $$Y = g(X)$$ where $g$ is a continuous function on $\mathbb R$. To find the expectation I can substitute the value of $Y$ to get $$\mathbb E[XY] = \mathbb E[Xg(X)]$$ but to determine the expectation of that I need to know the pdf of $Xg(X)$, how can I determine that?

Answer 1

No, you don't, since in general,

$$\mathbb E(h(X))=\int_{-\infty}^\infty h(x)f(x) dx$$

Now, let $h(X)=Xg(X)$, you get

$$\mathbb E(Xg(X))=\int_{-\infty}^\infty xg(x)f(x) dx$$