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An exponential distribution of rate $\lambda >0$ has density $f_X(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ and zero for $x<0$, with survival probability $\mathbb{P}(X>x)=e^{-\lambda x}$. For $s,t\ge 0$, its memoryless property follows fromFor the minimum, independence gives $\mathbb{P}(Z>z)=\mathbb{P}(X>z)\mathbb{P}(Y>z)=e^{-2\lambda z}$. HenceThe density is zero for negative $z$. The comparison probability follows from identical continuous laws and zero tie probability; directly it is ${\int }_0^{\infty }\lambda e^{-2\lambda y}dy=1/2$. These are competing exponential clocks.

For the last density, substitute $y=s^2$ into the normalizing integral to obtain $C=2{\int }_0^{\infty }{e}^{-s^2}ds=\sqrt{\pi }$, the Gaussian integral. Equivalently, each variable has a gamma distribution with shape $1/2$ and rate one. Independence permits convolution of the densities. For $z>0$,Putting $y=z{\sin }^2\theta$ turns the integral into $2{\int }_0^{\pi /2}d\theta =\pi$. ConsequentlyThis is also the additivity of independent gamma distributions with a common rate: their shapes add to one. Density values at the single endpoint zero are immaterial.

Initially, the infection and recovery transition intensities are $\beta N$ and $\gamma$. By competing exponential clocks, the first-event waiting time has distribution

The first event causes extinction precisely when it is recovery of the sole initially infectious individual. An infection would instead give two infectious individuals. The competing exponential clocks calculation therefore gives the first-event extinction probability

The competing exponential clocks calculation givesso $V$ is exponential with rate ${\lambda }_1+{\lambda }_2$. Also

Put $Q={\sum }_{j=1}^n{q}_j$. Since the variables have continuous distributions, the minimum is unique almost surely. Integrating over its possible value givesThis is the joint tail calculation for competing exponential clocks.

The two event clocks have independent exponential distributions with rates $\mu$ and $\gamma$. The competing exponential clocks identity gives