Following the question asked here : Express the event ${W≤w,M= 1,O≤t}$ in terms of the random variables $X$ and $Y$
I recall the notation $X, Y \sim \exp (\lambda)$, $W=\min (X, Y)$, $Z=\max (X, Y)$, $O=Z-W$, and $M=1_{X \leq Y}$.
I am asked to prove that $W$ and $(M,O)$ are independent.
I have been given the Hint You may assume without proof that it is enough to check that $P((W\in A, (M,O)\in B )= P((W\in A) P( (M,0) ∈ B)$, for $A= [0,w] and B={m}×[0,t]$, for all $w\ge0$, $m ∈ \{0,1\}$ and $t\ge0$
However, it appears this hint considers the same events- $(W\le w, M=1, O\le t)$ and $(W\le w, M=0, O\le t)$ as discussed in the link, which is calculated via a double integral, which can of course be split and computed as $2$ separate integrals multiplied together, immediately implying the proof (see hint), so I'm not sure why this question would be asked (by the lecturer), after asking the previous question, which is:
Find $(W\le w, M=1, O\le t)$ and $(W\le w, M=0, O\le t)$ (see the referenced post).
Any help would be much appreciated
Thankyou