Let $X\_1$ and $X\_2$ be i.i.d. normal. 

**The question is:** 
Can we find the joint distribution of a pair 
\begin{align}(U\_1,U\_2)=(\max(X\_1,X\_2), \min(X\_1,X\_2)).
\end{align} 

**What I did** 

Note that this is just the ordering of $X\_1$ and $X\_2$. In other words,
If $X\_1>X\_2$ then $(U\_1,U\_2)=(X\_1,X\_2)$ and if $X\_2<X\_1$ then $(U\_1,U\_2)=(X\_2,X\_1)$.

Then for every set $A$ we have that 
\begin{align}
P( (U\_1,U\_2) \in A)&= P( (U\_1,U\_2) \in A \mid X\_1>X\_2) P( X\_1>X\_2)+ P( (U\_1,U\_2) \in A \mid X\_1\le X\_2) P( X\_1 \le X\_2)\\
&= \frac{1}{2} P( (U\_1,U\_2) \in A \mid X\_1>X\_2)+ \frac{1}{2} P( (U\_1,U\_2) \in A \mid X\_1 \le X\_2)\\
&=P( (X\_1,X\_2) \in A \mid X\_1>X\_2)+ \frac{1}{2} P( (X\_2,X\_1) \in A \mid X\_1 \le X\_2)
\end{align}

Can somehow finish this?