Q:

What is the following expectation?

What is $E[Y_1I(x<Y_2,Y_2>Y_1)]$, where $Y_1$ and $Y_2$ are non-negative continuous random variables and x is a constant? $I(.)$ is the indicator function. Can this be written as follows?
\begin{align*}
E[Y_1I(x<Y_2,Y_2>Y_1)]&=\int_{0}^{\infty}yP(Y_2>x,Y_2>y)dG(y)\\
&=\int_{0}^{\infty}yP(Y_2>max(x,y))dG(y)\\
&=\int_{0}^{\infty}y\bar{G}(max(x,y))dG(y)
\end{align*}

A:

It's correct. Another way would be: $$\begin{align}\mathbb E[Y_1\mathcal I(x<Y_2, Y_1<Y_2)]&=\int_0^\infty\int_{\max(x,y_1)}^\infty y_1 f(y_1)f(y_2)dy_2dy_1\\&=\int_0^\infty y_1 (1-G(\max(x,y_1)))f(y_1)dy_1\\&=\int_0^\infty y \bar G(\max(x,y))dG(y)\end{align}$$
where $\bar G$ denotes the complementary CDF.

