Conditional split of two exponential times ($E[S_1\mid S_1+S_2=t]=\frac{1}{{\lambda }_1-{\lambda }_2}-\frac{t}{e^{({\lambda }_1-{\lambda }_2)t}-1}$)

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 $t/2$.

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