Originally Posted by

**Laurent** There is a mistake in the exponents: it is $\displaystyle e^{-\lambda_1 r}$ when $\displaystyle r>0$ and $\displaystyle e^{-\lambda_2 r}$ else.

How to find the result with little computation (but some nontrivial knowledge): first, the probability that $\displaystyle T_1>T_2$ is $\displaystyle \frac{\lambda_2}{\lambda_1+\lambda_2}$ (classic useful computation); then, by the memoryless property of exponential distribution, conditionally to $\displaystyle T_2$ and to the event $\displaystyle \{T_1>T_2\}$, the distribution of $\displaystyle T_1-T_2$ is exponential with parameter $\displaystyle \lambda_1$ (note that since this distribution does not depend on $\displaystyle T_2$, it was unnecessary to condition by $\displaystyle T_2$). This gives the "$\displaystyle r>0$" part: $\displaystyle \frac{\lambda_2}{\lambda_1+\lambda_2}\lambda_1 e^{-\lambda_1 r}$, and symmetry gives the other part.