Let $X \sim \exp(1)$, $Y \sim \exp(1)$ $Z=X-Y$.

$X,Y$ are independent.

What is the distribution of Z?

For $t\geq0$, I simply calculated that the old fashioned way.

$F\_Y(t)=P(Z\leq t)=P(X-Y\leq t)=\int\_{0}^{\infty}\int\_{0}^{t+y}e^{-x}e^{-y}dxdy=1-\frac{1}{2}e^{-t}$

What can I do about the second case? $t<0$? 

If the joint PDF of $X$ and $Y$ is non-negative, how can I integrate in this region?