Maximum of standard normals

Question

Let $X_1,...,X_n$ be independent standard normals. Let $Z=\max_iX_i$. Then

$F_Z(z)=F_X(z)^n$ and $f_Z(z)=nF_X(z)^{n-1}f_X(z)$

Now let's take the expectation of $Z$.

$E(Z)=\int_{-\infty}^{\infty}znF_X(z)^{n-1}f_X(z)dz$

Since $F_X(z)^{n-1}\leq1$ for every $z$, we can bound this as

$E(Z)=\int_{-\infty}^{\infty}znF_X(z)^{n-1}f_X(z)dz\leq \int_{-\infty}^{\infty}znf_X(z)dz=0.$

So I get that $E(Z)\leq 0$ which makes no sense, but I cannot figure out where is my mistake.

Answer 1

You cannot bound the expectation by substituting 1 for $F_X(z)^{n-1}$, because not all the values in the integral are positive hence, it isn't guaranteed to be bigger than expectation.

Intuitively, by substituting 1, you are effectively giving more weight to negative values than they have in expectation, which should result in a lower than expected value.