Let $X, Y$ be independent random variables with distribution functions $F_{X}, F_{Y}$. Show that $U=\min \{X, Y\}, V=\max \{X, Y\}$ have distribution functions

$$F_{U}(u)=1-\left(1-F_{X}(u)\right)\left(1-F_{Y}(u)\right), \quad F_{V}(v)=F_{X}(v) F_{Y}(v)$$

Now let $X, Y$ be independent random variables, each having the exponential distribution with parameter 1. Show that $U$ has the exponential distribution with parameter 2 , and that $V-U$ is independent of $U$.

Hence or otherwise show that $V$ has the same distribution as $X+\frac{1}{2} Y$, and deduce the mean and variance of $V$.

[You may use without proof that $X$ has mean 1 and variance 1.]