# Maximum value of n Normal distribution with different means

I have $n$ different Normal Distribution $N_i = \mathcal{N}(\mu=\mu_i, \sigma=\sigma^*)$. I would like to know what is the distribution of $\max_i N_i$ Is there some paper where this result is shown?

• Given $n$ independent random variables $X_1, \ldots, X_n$, we have $$\verb/Prob/\left[\max\limits_{1 \le i \le n} X_i \le x\right] = \verb/Prob/\left[ X_1 \le x \land \cdots \land X_n \le x \right] = \prod_{i=1}^n \verb/Prob/[ X_i \le x ]$$ CDF of max is just product of individual CDFs. – achille hui Nov 12 '16 at 11:42
• Can't I write this product as a new normal distribution? – Samuele Nov 12 '16 at 11:50
• You cannot, max of normal random variables is not normal distributed – achille hui Nov 12 '16 at 11:57