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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.

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Not the entire answer, but here's an observation.

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 $PDF_Y$.

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