Cumulative distribution function


definiendum $F\in\mathrm{CDF} $
inclusion $F:\mathbb R\to\mathbb R$ … right-continuous function, monotonically increasing
postulate $\lim_{x\to\ -\infty} F(x)=0$
postulate $\lim_{x\to\ +\infty} F(x)=1$

Let $S:({\mathbb D}\to {\mathbb R}_{\ge 0})\to{\mathbb R}_{\ge 0}$ be linear.
If $f$ with $\infty>Sf>0$, then $\bar{f}:=\frac{1}{Sf}\cdot f$ has $S\bar{f}=\frac{Sf}{Sf}=1$.
So we can use such $S$ to normalize functions.

For ${\mathbb D}={\mathbb N}$ the general case is $Sf:=\sum_{n=0}^\infty (L_nf)(n)$, where $(L_n)$ is a suitable sequence of linear operations (e.g. differential operators). For $L_n={\mathrm{id}}$ we get the standard sum (see below).
For ${\mathbb D}\subseteq{\mathbb R}^m$ we have integrals.

Let $a:{\mathbb N}\to{\mathbb R}_{\ge 0}$ be a sequence, then

$\bar{a}:{\mathbb N}\to[0,1]$

$\bar{a}(n):=\frac{1}{\sum_{k=0}^\infty a(k)}\cdot a(n)$

$\sum_{n=0}^\infty \bar{a}(n)=1$
The “monomial bump” on $[-d,d]$, which goes against the constant probability $\frac{1}{2d}$ for large $n$:
$\int_{-d}^dP_{n,d}(x)\,{\mathrm d}x=1$
Die Funktion hängt mit dem sog. Epanechnikov-Kern zusammen.


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