Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ from $x$ and $y$ respectively.

What is the probability that $\mbox{argmax}\ X = \mbox{argmax}\ Y$.

Alternatively, suppose that $x_1$ is maximal element, what is the probability that $y_1$ is maximal too.

Any references pointing to the solution of either question would also be appreciated.


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