# Distribution of largest sample from normal distribution.

Given $n$ independent random variables $X_i$ with normal distribution, mean $\mu$, variance $\sigma^2$, what is the distribution of $\max\limits_{i=1}^n(X_i)$ ?

In particular I am interested in whether it would be Normal or close to Normal as well, and what the mean and variance would be.

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The random variable $Z = \max_{i=1}^n(X_i)$ is known as order statistics, and is sometimes denoted as $X_{n:n}$.

The cumulative density function of $Z$ is easy to find: $$F_Z(z) = \mathbb{P}\left(Z \leqslant z\right) = \mathbb{P}\left(\max_{i=1}^n(X_i) \leqslant z\right) = \mathbb{P}\left( X_1 \leqslant z, X_2 \leqslant z, \ldots, X_n \leqslant z\right)$$ using independence: $$F_Z(z) = \left(F_X(z)\right)^n$$ Thus the density function is $$f_Z(z) = n f_X(z) F_X^{n-1}(z)$$ In particular, it follows that $Z$ is not normal.

Expected values of $Z$ are known in closed form for $n=1,2,3,4,5$ (asking Mathematica):

In[31]:= Table[
Mean[OrderDistribution[{NormalDistribution[m, s], n}, n]], {n, 1, 5}]

Out[31]= {m, m + s/Sqrt[Pi], m + (3 s)/(2 Sqrt[Pi]),
m + (6 s ArcTan[Sqrt[2]])/Pi^(3/2),
m - (5 s)/(2 Sqrt[Pi]) + (15 s ArcTan[Sqrt[2]])/Pi^(3/2)}

Large $n$ asymptotics is discussed here.

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