Conditional distribution of $U$ given $\max(U,V)$ for $U$, $V$ i.i.d. uniform on $(0,1)$

Question

Suppose $U,V$ are i.i.d. random variables following Unif$(0,1)$, what is the conditional distribution of $U$ given $Z:=\max(U,V)$ ?

I tried writing $Z=\Bbb{I}\cdot V+(1-\Bbb{I})\cdot U$ where $\Bbb{I}=\begin{cases}1&U<V\\0&U>V\end{cases}$

But I am not getting anywhere.

Answer 1

Conditional CDF of $U$ given $Z$ is:

\begin{eqnarray*} F_{U|Z}(u|z) & = & \Pr(U\leq u|Z =z) \\ &= & \Pr(U\leq u|\max(U, V) =z) \\ &= & \Pr(U\leq u, U \leq V|\max(U, V) =z) + \Pr(U\leq u, U > V|\max(U, V) =z) \\ &=& \Pr(U\leq u|U\leq V, \max(U, V) =z)\Pr(U\leq V|\max(U, V) =z) \\ && + \Pr(U\leq u|U > V, \max(U, V) =z)\Pr(U > V|\max(U, V) =z) \\ &=& \Pr(U\leq u|U\leq V, V =z)\times \frac{1}{2} + \Pr(U\leq u|U > V, U =z)\times \frac{1}{2} \\ &=& \begin{cases} 0 & if \ u\leq 0 \\ \displaystyle\frac{u}{2z} & if \ 0 < u < z < 1 \\ 1 & if \ u\geq z \end{cases}\end{eqnarray*}