# Maximum of Correlated random variables

I am trying to find the CDF $Z = \max(X_1,X_2,\dots,X_N)$ and in my case $X_i$ are correlated. Is there any transfer domain or one to one function where I can derive the CDF and invert back to current domain ? Or how to handle this type of problem ?

Edit: Since the correlated joint distribution is not known ( but marginal distribution is known), I was wandering if we can transfer the variables in other domain e.g. multiplying with another matrix (whitening) so that the joint distribution would be simple multiplication of marginal distribution. But I am not sure about the equivalent one to one transforming function for maximum.

• $$F_Z(z) = P\{Z \leq z\} = P\{X_1\leq z, X_2\leq z, \cdots, X_N\leq z\} = F_{X_1,X_2,\cdots, X_N}(z,z,\cdots, z)$$ where just knowing the correlations between the $X_i$ is not enough to determine the right hand side. Commented Oct 23, 2014 at 16:37
• I changed \text{max} to \max. This is standard usage and not only prevents italicization, but also does some things that \text{max} does not do: (1) It results in proper spacing in expressions like $a\max b$; and (2) In a "displayed" rather than "inline" context, it affect the positions of subscripts in things like $\displaystyle\max_{a\in A} f(a)$. ${}\qquad{}$ Commented Oct 23, 2014 at 17:11
• You need more information about the joint distribution than what is given here. Commented Oct 23, 2014 at 17:12

If you know the joint distribution $F(\{X_i\leq a_i\})$, then in general:
If $Z:=\max\{X_i\}$ then $F(z)=F(\{x_i\leq z\})$