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### Probability random variable is less or equal to k-th out of two samples when ordered

If the distribution is continuous then WLOG we can assume that we are dealing with uniform distribution on $(0,1)$. Then for $u\in(0,1)$: P\left(X=X_{\left(r\right)}\text{ and }X=Y_{\left(s\right)}\...
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### Probability random variable is less or equal to k-th out of two samples when ordered

\begin{align} \mathbb{P}(\forall k\ X > X_ {(k)}, X > Y_ {(k)}) &= \mathbb E\left[\prod_{k=1}^n\mathbb I_{X > X_ {(k)}}\prod_{k=1}^n\mathbb I_{X > Y_ {(k)}}\right]\\ &= \mathbb E^X\...
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If the question was just about guessing the position of the first-seen value $x_1$ among the $n$ then it would not be too difficult: you would guess $r_1$ to be the value $\hat r_1$ which maximises ...