Is expected value $E[X_1;X_1\leq X_2]= E[X_1;X_1< X_2]$

Question

Let $X_1, X_2$ two independent random variables with PDF $f_{X_i}(x_i)$. Is this formula is true $$E[X_1;X_1\leq X_2]= \int_{x_2=0}^{\infty}\Big(\int_{x_1=0}^{x_2}x_1 f_{X_1}(x_1)d x_1\Big)f_{X_2}(x_2)dx_2$$

I am asking if the expected value $$E[X_1;X_1\leq X_2]= E[X_1;X_1< X_2]$$,

or $$E[X_1;X_1< X_2]=E[X_1]-E[X_1;X_1\leq X_2].$$

Thanks

Answer 1

If $X_1$ and $X_2$ are independent random variables with a densities then $P\{X_1=X_2\}=0$ so it is true that $E(X_1;X_1 \leq X_2)=E(X_1;X_1 < X_2)$.