How to calculate $P (Y > X+1)$ Let the joint probability distribution of $X$ and $Y$ is given by
$$f(x,y)=\left\{ \begin{align}
  & {{e}^{-(x+y)}}\quad,x>0\,\,,\,\,y>0 \\ 
 & 0\qquad\quad\,\,\,,\operatorname{o}.w \\ 
\end{align} \right.$$
I want to compute $P (Y > X+1)$ in the region $D$.


It was very  nice! 
For dummies: We can separate terms for the exponential function, namely,
$\int_{0}^{+\infty} \Big(\int_{1+x}^{+\infty} e^{-y} dy\Big)e^{-x}dx=\frac{1}{2e}.$ 
