Problem with two correlated random normals Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal variables with a correlation of $r$.) You take $N$ samples from this variable, such that you have a $N \times 2$ matrix - the first column contains the samples from the first dimension, the second column contains the samples from the second dimension. (These columns, of course, have a correlation of $r$.)
Here comes the crucial part. You take the $K$ samples which are highest from the first column. Then, you choose the number from among those $K$ which is highest in the second column.
What is the distribution of this final number? It's easy to simulate, but is it analytically solvable (or does anyone know where I could start looking)?
 A: I don't have a full solution, but I'd start with the following. Your two sequences of random numbers $x_{1i},x_{2i}$ can be generated by applying cholesky decomposition to the correlation matrix $\Sigma$, then multiplying the cholesky matrix by two independent randoms $\xi_{1i},\xi_{2i}$:
$$x_{1i}=\xi_{1i}\\
x_{2i}=\xi_{1i}+\sqrt{1-\rho^2}\xi_{2i}$$
Now, when you take the largest K numbers $i_j,j=1,2,\dots,K$ of the first sequence $x_{1i_j}=\xi_{1i_j}$, you also pick the second sequence's subset $x_{2i_j}$. What is the distribution of $x_{2i_j}$?
Note, that the distribution of $\xi_{2i_j}$ is independent from $x_1$, therefore it must be still normal. So, if you know the distribution of $\xi_{1i_j}$ then the distribution in question is of the sum of independent random numbers one of which is definitely normal.
The set of K largest $\xi_{1i_j}$ is a set of K order statistics of normal distribution sample of numbers. I need to think of what would be the distribution, it's definitely NOT normal.
