This is an extension of my previous question from the post

iid and correlated order statistics a comparison

Consider the two random variables $X_1$ and $X_2$ defined via non-negative random variables $Y_1$ and $Y_2$ as

$ X_1=\min(Y_1,Y_2)\\$ where $Y_1$ and $Y_2$ are i.i.d and

$ X_2=\min(Y_1,Y_2)$ - where $Y_1$ and $Y_2$ are correlated.

The question is can we say $X_1 \leq X_2$ always? How to prove it?

  • $\begingroup$ Your X's are defined on different probability spaces; hence, they cannot be compared as formulated. Can you describe how the $Y$ pairs for $X_1$ are related to the $Y$ pairs for $X_2$? $\endgroup$ – user76844 Jan 15 '15 at 4:11

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