Prove $E[g(X,Y)|X\geq a, Y\geq b]=E[g(X+a,Y+b)]$

Question

Prove $E[g(X,Y)|X\geq a, Y\geq b]=E[g(X+a,Y+b)]$ where $X,Y$ are independent exponential random variables with common rate $\lambda >0$ and $g(x,y)$ is a continuous function of $x$ and $y$. Let $a\geq 0$ and $b\geq 0$. I'm trying to do this: $$E[g(X,Y)|X\geq a,Y\geq b]=\int_a^\infty\int_b^\infty g(x,y)\cdot f_{X,Y|X\geq a,Y\geq b}(x,y)\space dy \space dx$$ I think that $$f_{X,Y|X\geq a,Y\geq b}(x,y)=\frac{f_{X,Y}(x,y)}{P(X\geq a)P(Y\geq b)}$$ but I do not know the joint pdf of X and Y. (I also split up the probability on the bottom since X and Y are independent). Are there any suggestions for what I can do from here?

Answer 1

Hint 1: Recall that if $X$ and $Y$ are independent then $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ so you actually know what the joint density is.

Hint 2: Consider the change of variables $(x,y)=(u+a, y+b)$ for the integration.

Let me know if this did not help.