Verteilungsfunktion

A function $F$ $F:\mathbb{R}\to [0,1]$ is called CDF when ${\lim }_{x\to -\infty }F(x)=0{\lim }_{x\to +\infty }F(x)=1$ and $F$ is

• montonocally increasing (Proof Monotonic Increase)
• right continuous (Proof of Right Continuity)

$F(x):=P((-\infty ,x])$

Properties of CDF

• Left limits exist
• jump positions are countable
• If the ECDF converges uniformly to the CDF then all interval frequencies converge and the limit is called interval probability
• for random numbers from a CDF the ECDF wil converge uniformly almost surely (Glivenko-Cantelli)

Relation to PDF

If $P$ is defined by PDF $f$, one can write $F(x)={\int }_{-\infty }^{x}f(y)dyx\in \mathbb{R}$

Aus dem Hauptsatz der Differential- und Integralrechnung folgt also wenn $f$ stetig ist, der Zusammenhang: $F^{\prime }=f$ This way we also know that $f(t)={\lim }_{h\to 0}\frac{F(t+h)-F(t)}{h}$ which can be used to give a formula for a small interval probability based on $f(t)$ > $\mathbb{P}((t,t+h])\approx f(t)h$ for small $h$.

Important Classes of CDF

• piecewise constant CDF
  • for example of uniformly discrete item
  • can be described with PMF
• continous CDF
• Mixed types between continous and piecewise constant CDF
• CDF with PDF where the PDF is the slope of the CDF at a given continouity point
  • PDF times $h$ can approximate the interval probability of $(t,t+h]$ for $h$ small