Past exam of the probability course of the University of Oxford / 2014 / 2 / d

The Poisson interarrival times of a homogeneous Poisson process are independent and identically distributed with the exponential distribution of rate $\lambda$. Consequently $Y_1,Y_2$ are independent random variables and their joint probability density isHere $Y_2$ is the second interarrival duration, rather than the time of the second arrival.

The inverse change of variables is $y_1=uv$, $y_2=(1-u)v$, on $0<u<1$, $v>0$. Its absolute Jacobian determinant isThe change of variables for probability densities therefore givesThis factorizes on a product domain into a unit uniform density in $u$ and a shape-two Gamma density in $v$. Thus $U$ and $V$ are independent, with $U\sim \mathrm{Uniform}(0,1)$ and $V\sim \mathrm{Gamma}(2,\lambda )$. This is the shape-$(1,1)$ case of beta-gamma independence.