Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 11F (Probability)

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

The exponential memoryless property gives

After the first of two equal-rate clocks rings, the residual lifetime of the other clock is, by memorylessness, a fresh exponential variable of rate $\lambda$, independent of the first ringing time and of which clock rang. Henceand these variables are independent.

Take the spatial Fourier transform of the wave equation. The transformed initial-value problem isand hence

Part (c) and the convolution theorem turn the first term intoFor the second term, choose $p$ with $p^{\prime }=g$. Part (b) says $\tilde{g}=ik\tilde{p}$, soPart (c) now makes its inverse transformCombining the two terms gives the D'Alembert formula

Let $A=2V$ and $B=W-V$. Part (c) makes $A$ and $B$ independent identically distributed exponential variables of rate $\lambda$. The change of variables $(A,B)=(rs,(1-r)s)$ has Jacobian $s$, and integrating ${\lambda }^{2}s{e}^{-\lambda s}$ over $s>0$ gives density one for $0<r<1$. Therefore the uniform ratio of independent exponential variables gives