# Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $$Y = \max(X_1, ..., X_n)$$ where $$X_i \sim \mathcal{N}(\mu_i, \sigma^2)$$.

Note that the $$X_i$$ are independent, but not identically distributed. That is, they have the same variance $$\sigma^2$$ but different means $$\mu_i$$.

The answer to my previous question may help: Distribution of maximum of normally distributed random variables.

For the $$\max$$ function, the CDF combines very easily. That is, the probability that $$y < \Lambda$$ equals the probability that all $$x_i < \Lambda$$. If furthermore the $$X$$ are independent then it is true that:
$$CDF_Y (\Lambda) = \prod_i^n \left[ CDF_{X_i}(\Lambda) \right]$$
At least formally you can suspect that this allows you to compute $$CDF_Y$$ and therefore $$PDF_Y$$.