TBan-I

Detour: Extreme Value Statistics

Consider the $M$ energies $E_1,\dots ,E_M$ as independent and identically distributed (i.i.d.) random variables drawn from a distribution $p(E)$. It is useful to introduce the cumulative probability of finding an energy smaller than E

$P(E)=\underset{-\mathrm{\infty }}{\overset{E}{\int }}dxp(x)$

We define:

$E_{\min }=\min (E_1,\dots ,E_M)$

Our goal is to compute the cumulative distribution:

$Q_M(E)\equiv \text{Prob}(E_{\min }>E)$

for large $M$. To achieve this, we rely on three key relations: