Let $X_1, X_2, X_3, X_4$ be independent and identically distributed continuous random variables. Then we know that for any $s,t$ real numbers $$P(X_1\leq s, X_2\leq t)=P(X_1\leq s)P(X_2\leq t)$$.
Then, can I claim that
- $P(X_1\leq X_3, X_2\leq X_3)=P(X_1\leq X_3)P(X_2\leq X_3)$?
If this is true, then I can also claim that $$P(\max(X_1,X_2)<\min(X_3,X_4))=P(X_1<\min(X_3,X_4))P(X_2<\min(X_3,X_4))$$ and then iterating this I can get $$P(\max(X_1,X_2)<\min(X_3,X_4))=P(X_1<X_3)^4$$.
This is not correct because my simulation result contradicts. What is wrong here?