Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables.
Let $X_1,…,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables.
Consider that $X_i$ and $U_j$ are independents, for all $i,j$.
Show that the event $\{U_1 > U_2 > U_3\}$ is independent of the $(U_{(3)},X_{(1)})$, where $U_{(3)} = \max\{U_1,U_2,U_3\}$ and $X_{(1)} = \min\{X_1,…,X_n\}$.
I have no idea to start. How'd be the definition of independence between events and random variables?
Best Answer
An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.
In short you need to establish whether: $${\forall u\in(0;1)~\forall x\in(0;1):\\\quad\mathsf P(\{U_1{>}U_2{>}U_3\})=\mathsf P(\{U_1{>}U_2{>}U_3\}\mid u{=}\max\{U_i\}_{i=1}^3,x{=}\min\{X_j\}_{j=1}^n)}$$