Competing exponential clocks

For independent exponential variables $S_i$ of rates $q_i$ and $Q={\sum }_i{q}_i$, let $T={\min }_i{S}_i$ and let $K$ be the minimizing index. ThenConsequently $T$ is exponential with rate $Q$, $\mathbb{P}(K=k)=q_k/Q$, and $K$ and $T$ are independent.

Let $S,T$ be independent random variables with exponential distributions of respective positive rates $\alpha ,\beta$. Condition on $T=t$ and use the survival function of $S$:At $u=0$ this is $P(S>T)$. Dividing by that probability proves that the positive difference conditional on $S>T$ again has an exponential distribution with rate $\alpha$. The memorylessness of the exponential distribution persists after an independent competing event has already occurred. Replacing the difference by a negative sentinel when $S\le T$ adds an atom of a measure of mass $\alpha /(\alpha +\beta )$ at that sentinel without changing the positive tail.