= Positive difference of independent exponential waiting times
{title2=$P(S-T>u)=\frac{\beta}{\alpha+\beta}e^{-\alpha u},\quad u\geq0$}
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 $S$:
$$
P(S-T>u)=\int_0^\infty\beta e^{-\beta t}e^{-\alpha(t+u)}dt
=\frac{\beta}{\alpha+\beta}e^{-\alpha u},\qquad u\geq0.
$$
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 $S\leq T$ adds an <atom of a measure> of mass $\alpha/(\alpha+\beta)$ at that sentinel without changing the positive tail.
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