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

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.

• If it helps: \begin{align*}\mathbb{P}(U \leq u, Z \leq z) &= \mathbb{P}(U \leq u, U \leq z, V \leq z) \\&= \mathbb{P}(U \leq \min(u,z), V \leq z) \\&= \mathbb{P}(U \leq \min(u,z)) \, \mathbb{P}(V \leq z)\end{align*}
– Tom
Aug 27, 2016 at 20:25
• @Tom SO, $$=\min (u,z)\cdot z$$? Aug 27, 2016 at 21:26
• For $u, z \in (0,1)$ this seems correct to me, unless I'm missing something. This isn't the conditional distribution, of course, just the joint CDF.
– Tom
Aug 27, 2016 at 21:27
• The conditional distribution of $U$ conditionally on $Z=z$ is the measure $\frac12\mathrm{Unif}(0,z)+\frac12\delta_z$, that is, $P(U=z\mid Z=z)=\frac12$ and, for every $u$ in $(0,z)$, $P(U\leqslant u\mid Z=z)=\frac{u}{2z}$.
– Did
Aug 28, 2016 at 18:50
• Cross-posted at stats.stackexchange.com/questions/232085/…. Jan 4, 2020 at 6:50

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