Verteilungsfunktion

A function FF

F:R→[0,1]F:\mathbb{R}\rightarrow [0,1]

is called CDF when

lim⁡x→−∞F(x)=0lim⁡x→+∞F(x)=1\lim _{x \rightarrow-\infty} F(x)=0 \quad \lim _{x \rightarrow+\infty} F(x)=1

and FF is

F(x):=P((−∞,x])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 PP is defined by PDF ff, one can write
F(x)=∫−∞xf(y)dyx∈RF(x)=\int_{-\infty}^{x} f(y) d y \quad x \in \mathbb{R}

Aus dem Hauptsatz der Differential- und Integralrechnung folgt also wenn ff stetig ist, der Zusammenhang:
F′=fF' = f
This way we also know that
f(t)=lim⁡h→0F(t+h)−F(t)hf(t)=\lim _{h \rightarrow 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)f(t) > P((t,t+h])≈f(t)h\mathbb{P}((t, t+h]) \approx f(t) h
for small hh.

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 hh can approximate the interval probability of (t,t+h](t,t+h] for hh small