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:

Failed to parse (syntax error): {\displaystyle \text{Prob}[n(x) = k] = M \binom{M-1}{k}} }

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

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