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. 



Probability that maximal elements has the same position in samples from correlated random variables
The question can apparently be clarified as follows: 



> 
> Let $(X\_1,Y\_1),\dots,(X\_n,Y\_n)$ be iid pairs of real-valued random variables with a common continuous joint cumulative distribution function (cdf) $F$, so that $F(x,y)=P(X\_i<x,Y\_i<y)$ for all $i$ and all real $x,y$. What is then the probability 
>  \begin{equation}
>  p:=P(\exists i\ \forall j\ne i\ X\_i>X\_j, Y\_i>Y\_j)? 
> \end{equation}
> 
> 
> 


To answer this question, note first that the random pairs $(X\_1,Y\_1),\dots,(X\_n,Y\_n)$ are exchangeable and also, because of the continuity of $F$, $P(X\_i=X\_j)=0=P(Y\_i=Y\_j)$ for all distinct $i$ and $j$. So,  

\begin{equation}
 p=np\_1,\quad p\_1:=P(\forall j\ne 1\ X\_1>X\_j, Y\_1>Y\_j). 
\end{equation}
Next, 
\begin{equation}
 p\_1=Eg(X\_1,Y\_1),\quad g(X\_1,Y\_1):=P(\forall j\ne 1\ X\_1>X\_j, Y\_1>Y\_j|X\_1,Y\_1). 
\end{equation}
Further, for all real $x,y$
\begin{equation}
 g(x,y)=P(\forall j\ne 1\ x>X\_j, y>Y\_j)=F(x,y)^{n-1}. 
\end{equation}
Thus, 
\begin{equation}
 p\_1=Eg(X\_1,Y\_1)=\int\_{\mathbb R^2}F(x,y)^{n-1}\,dF(x,y)
\end{equation}
and 
\begin{equation}
 p=n\int\_{\mathbb R^2}F(x,y)^{n-1}\,dF(x,y). 
\end{equation}




---


In particular, if $X\_1$ and $Y\_1$ are independent, so that $F(x,y)=G(x)H(y)$ for all real $x,y$ (where $G$ and $H$ are the cdf's of $X\_1$ and $Y\_1$), then 
\begin{multline}
 p=n\int\_{\mathbb R^2}G(x)^{n-1}H(y)^{n-1}\,dG(x)\,dH(y) \\ 
 =n\int\_{\mathbb R}G(x)^{n-1}\,dG(x)\;
 \int\_{\mathbb R}H(y)^{n-1}\,dH(y) \\ 
 =n\int\_0^1 u^{n-1}\,du\;\int\_0^1v^{n-1}\,dv =\frac1n,
\end{multline}
which also follows immediately because the random pairs $(X\_1,Y\_1),\dots,(X\_n,Y\_n)$ are exchangeable. 


