= Independent minimum and gap of two exponential variables
{title2=$(U,V-U)\sim\operatorname{Exp}(2\lambda)\otimes\operatorname{Exp}(\lambda)$}
For two independent rate-$\lambda>0$ <exponential distributions>, set $U=\min(X,Y)$ and $V=\max(X,Y)$. Summing the two ordering branches gives <joint probability density> $2\lambda^2e^{-\lambda(2u+d)}$ for $(U,d=V-U)$ with $u,d>0$. It factors into independent exponential <probability density functions>. Thus $V$ is distributed as $X+Y/2$, with <expected value> $3/(2\lambda)$ and <variance> $5/(4\lambda^2)$. This spacing result is the order-statistic manifestation of the <memorylessness of the exponential distribution>.
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