There are two index sets $I= \{1, 2, \dots, m\}$ and $J = \{1, 2, \dots, n\}$. Then, I have independent random variables $X_{ij}, \forall i \in I, j \in J$. Fix $i \in I$; then we have $X_{ij}$ is identical for all $j \in J$.
Fix $k \in J$, I am wondering if I could have the following probability inequality$$ P\left(X_{1k} \leq Y, X_{2k} \leq Y \mid Y = \max_{I\in I} \min_{j \in J} X_{ij}\right) \leq $$ $$ P\left(X_{1k} \leq Y \mid Y = \max_{I\in I} \min_{j \in J} X_{ij}\right) \cdot P\left( X_{2k} \leq Y \mid Y = \max_{I\in I} \min_{j \in J} X_{ij}\right)$$
If the inequality holds, how to show it? If the inequality does not hold, can we have some inequality of a similar format? Thank you very much!
