Hoping someone with more smarts than myself can let me know if I am doing this right.
The solution I arrive at is $-\infty$ and I am unsure if that makes any sense at all. Grateful for any help.
Question
A joint density function is given by
$f(x,y) = \begin{cases} e^{-x-y} & \text {for $x\ge0, y\ge0$} \\ \\ 0 & \text{otherwise} \\ \end{cases} $
Find $ P(x+y\le2)$
The integral I came up with is
$ \int_{0}^{\infty}\int_{0}^{2-y} e^{-x-y} \,dx dy$
which evaluates as
$ \int_{0}^{2-y} e^{-x-y} \,dx =e^{-y}-\frac{1}{e^2}$
and
$ \int_{0}^{\infty}(e^{-y}-\frac{1}{e^2})\,dy =-\infty-(-1)$
So is $-\infty$ a valid answer or have i messed up somewhere?