CDF of maximum of $n$ correlated normal random variables The maximum of $n$ normal i.i.d. random variables
$$Y=\max\{x_1,...,x_n\},$$
$$x_i \sim N[0,1]$$
has the CDF
$$P(Y\le y)=\Phi(y)^n  $$
but how does the CDF look like, if the variables are identically correlated?
$$\text{corr}(x_i, x_j)=\rho,$$
$$0\le\rho\le1.$$
I am looking for the analytical formula
$$P(Y\le y)=f(y,n,\rho).$$

If $X_i$ are exchangeable multivariate Normal variables you can write
$X_i= Y_i +Z$
where $Y_i$ are independent Normal$(0,\sigma^2)$ and $Z$ is Normal$(0,\tau^2)$, independent of the $Y_i$ (and $\rho^2=\tau^2/(\tau^2+\sigma^2)$)
So, $\max X_i=\max(Y_i+Z)= Z+ \max Y_i$ and the distribution of the maximum is the convolution of the distribution of $Z$ and the distribution of $n$ independent Normals.
That is, the exponent doesn't change. The distribution is the sum of a part with exponent $n$ (that vanishes at $\rho=1$) and a part with exponent 1 (that vanishes at $\rho=0$)
If $f_z$ is the density of $z$ and $f_y$ is the density of of $\max Y_i$ (which you already know), the density $f_x$ of $\max X_i$ is
$$f_x(x)=\int_{-\infty}^\infty f_z(z)f_y(x-z)\,dz.$$
It probably doesn't have a simple closed form.
