= Conditional split of two exponential times
{title2=$E[S_1\mid S_1+S_2=t]=\frac1{\lambda_1-\lambda_2}-\frac t{e^{(\lambda_1-\lambda_2)t}-1}$}
For <independent> <exponential distributions> with rates $\lambda_1>\lambda_2>0$, let $T=S_1+S_2$. Dividing the joint density of $(S_1,T)$ by the <hypoexponential distribution> density of $T$ gives a <truncated exponential distribution> on $(0,t)$ with rate $\lambda_1-\lambda_2$. Its displayed <conditional expectation> supplies an E-step when only pair totals are observed. At equal rates the conditional split is uniform, with mean $t/2$.
// Target: probability-and-statistics.bigb
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