Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 7 / 5 / Solution

Divide the joint first-event/type-A density by the type probability. For $\alpha >0$,It integrates to one, so the conditional time is exponential with rate $\alpha +\beta$. Type-A events that win the competition tend to occur sooner than an unrestricted latent A time. In fact the same conditional density holds for B, so the observed first-event time and its type are independent in this constant-rate model.