===== Cumulative distribution function =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $F\in\mathrm{CDF} $ |
| @#AAFFAA: inclusion   | @#AAFFAA: $F:\mathbb R\to\mathbb R$ ... right-continuous function, monotonically increasing |
| @#55EE55: postulate   | @#55EE55: $\lim_{x\to\ -\infty} F(x)=0$ |
| @#55EE55: postulate   | @#55EE55: $\lim_{x\to\ +\infty} F(x)=1$ |

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>$P=F'$

>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
> 
>$\bar{a}:{\mathbb N}\to[0,1]$
>
>$\bar{a}(n):=\frac{1}{\sum_{k=0}^\infty a(k)}\cdot a(n)$
>
>has
>$\sum_{n=0}^\infty \bar{a}(n)=1$

>The "monomial bump" on $[-d,d]$, which goes against the constant probability $\frac{1}{2d}$ for large $n$:
>$P_{n,d}(x):=\frac{1}{2d}\left(1+\frac{1}{2n}\right)\cdot\left(1-\left(\frac{x}{d}\right)^{2n}\right)$
>$\int_{-d}^dP_{n,d}(x)\,{\mathrm d}x=1$
>Die Funktion hängt mit dem sog. Epanechnikov-Kern zusammen.

=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Cumulative_distribution_function|Cumulative distribution function]]

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=== Subset of ===
[[ℝ valued function]], [[Monotonically increasing function]], [[Right-continuous function]]
=== Related ===
[[Probability space]]