Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 9F (Probability)

Put $Q={\sum }_{j=1}^n{q}_j$. Since the variables have continuous distributions, the minimum is unique almost surely. Integrating over its possible value givesThis is the joint tail calculation for competing exponential clocks.

Setting $t=0$ in part (i) givesAlsosoPart (i) now factors aswhich proves that $K$ and $T$ are independent.

This is the unit-rate Erlang distribution. For completeness, use convolution. The result is clear for $n=1$. Ifthen independence and the unit exponential density give, for $x>0$,Induction therefore proves

Conditioning on $N$ and using the moment-generating function of a unit exponential variable,where the geometric series converges precisely when $\theta <p$. This is the moment-generating function of an exponential variable of rate $p$, so the geometric sum of exponential variables satisfies