Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 41 / 5 / a / Solution

The sum of the two independent exponential times has a hypoexponential distribution. By convolution of probability densities, for $t>0$ and $d={\lambda }_1-{\lambda }_2>0$,Thereforeand $g(t)=0$ for $t<0$. This is a nonnegative normalized probability density function. At the equal-rate boundary it tends to ${\lambda }^{2}t{e}^{-\lambda t}$, a shape-two Gamma distribution.