Hypoexponential distribution

A hypoexponential distribution is the distribution of a sum of independent variables with exponential distributions. For two positive unequal rates $\lambda ,\mu$, convolution of independent random variables gives the probability density functionIt is zero for negative $s$. Its equal-rate limit is the shape-two gamma distribution with rate $\lambda$.

Two sequential independent memoryless reaction waits have a hypoexponential distribution. Convolution gives $p(t)=k_1{k}_2(e^{-k_{1}t}-e^{-k_{2}t})/(k_2-k_1)$. Equal rates give the Erlang distribution $k^{2}t{e}^{-kt}$. A nonzero mode distinguishes this ideal two-event model from a single exponential wait, but does not chemically identify the events.

If the independent exponential random variables $T_i$ have rates $i\lambda$, thenwhere the $E_i$ are independent exponential random variables of rate $\lambda$. To see this, the first of $N$ independent exponential clocks rings at rate $N\lambda$; after it rings, the memoryless property leaves $N-1$ independent rate-$\lambda$ clocks. The successive order statistic gaps are therefore independent with rates $N\lambda ,(N-1)\lambda ,\dots ,\lambda$. Their sum has the same distribution as $\sum T_i$. In particular, for $x>0$, the tail at $Nx$ has logarithm $-\lambda Nx+o(N)$, since $1-(1-e^{-\lambda Nx}{)}^N\sim N{e}^{-\lambda Nx}$. The sum divided by $N$ has a large deviation principle with speed $N$ and rate function $\lambda x$ on $x\ge 0$, infinity otherwise.

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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