Say X and Y are two independent random variables with exponential density\begin{split} f_{X}(x) = a e^{-ax}\end{split} and \begin{split} f_{Y}(y) = b e^{-by}\end{split}, then what is the probability density function of Z=X-Y? I'm trying to slove this problem, but I have no sense of the integrating regions. How will they be?
I tried Shai's hint \begin{split} {\rm P}(X - Y \le z) = \int_0^\infty {{\rm P}(X - Y \le z|Y = \tau )f_{Y}(\tau ){\rm d}\tau } = \int_0^\infty {{\rm F_{X}}(z + \tau )f_{Y}(\tau ){\rm d}\tau } \end{split}
I obtained this \begin{split} {\rm P}(Z \le z) = F_{Z}(z) = 1 - \frac b {a+b} e^{-a z} \end{split} But it's not converage to 1 when z is infinite, what's wrong with my calculation?