Let $X=\underset{i=1,...,N}\max(X_i)$ and $Y=\underset{i=1,...,N}\max(Y_i)$, where each pair of random variables $X_i$ and $Y_i$ are correlated with a correlation coefficient $\rho$, and have a joint PDF $f_{X_i,Y_i}\left(x,y\right)$
I am trying to prove an equation in a paper which states that the conditional PDF of $X=\underset{i=1,...,N}\max(X_i)$ given $Y=\underset{i=1,...,N}\max(Y_i)$; i.e., $f_{X|Y}(x|y)$ is written in terms of $f_{X_i,Y_i}\left(x,y\right)$ and the marginal PDF $f_{Y_i}(y)$, for the case of i.i.d $X_i$ and also $Y_i$ as follows:
\begin{equation} f_{X|Y}(x|y)=\frac{f_{X_i,Y_i}\left(x,y\right)}{f_{Y_i}(y)} \end{equation}
I tried to prove it but I end up having this ratio but multiplied with another one involving the joint CDF of $X_i,Y_i$ and the CDF of $Y_i$,
I would appreciate your support,
Thank you,