TBan-I

Detour: Extreme Value Statistics

Definition of $n(x)$:

Given a realization, $n(x)$ is defined as the number of random variables above the minimum $E_{\min }$ such that their value is smaller than $E_{\min }+x$. This quantity is a random variable, and we are interested in its average value:

$\overline{n(x)}=\sum _{k}k\text{Prob}[n(x)=k]$

The key relation for this quantity is:

$\text{Prob}[n(x)=k]=M(\genfrac{}{}{0ex}{}{M-1}{k})$

We use the following identity to sum over $k$:

$\sum _{k=0}^{M-1}k(\genfrac{}{}{0ex}{}{M-1}{k})(A-B{)}^k{B}^{M-1-k}=(A-B)\frac{d}{dA}\sum _{k=0}^{M-1}(\genfrac{}{}{0ex}{}{M-1}{k})(A-B{)}^k{B}^{M-1-k}=(M-1)(A-B)A^{M-2}$