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 $D$.


You just have to compute the integral of $e^{-(x+y)}$ over the given region, i.e.
$$ I = \int_{0}^{+\infty}\int_{1+x}^{+\infty}e^{-(x+y)}\,dy\,dx =\int_{0}^{+\infty}e^{-1-2x}\,dx=\color{red}{\frac{1}{2e}}.$$
