Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 2 / 10F / Solution

The inverse change of variables is $x=u$, $y=(v-u)/a$, with absolute Jacobian determinant $1/a$. Because the original random variables are independent, their joint density is $f(x)g(y)$. Thus the transformed joint density isThe support is important: $X,Y\ge 0$ implies $V\ge U\ge 0$.

For independent exponential random variables of rate $\lambda >0$ and $a=1/2$, this becomes $2{\lambda }^2{e}^{-2\lambda v+\lambda u}$ on $0\le u\le v$. Integrating out $u$ gives the density of the sum:It vanishes for $v<0$. On the other hand, the cumulative distribution function of the maximum isDifferentiating yields exactly the same density. The maximum and $X+Y/2$ have the same distribution, as described by maximum of two exponentials as a scaled sum; this does not assert that they are equal for the same pair of sampled values.