Dirac distribution

The Dirac distribution , sometimes also called point distribution , deterministic distribution , unit mass or degenerate distribution , is a special probability distribution in stochastics. It is one of the discrete probability distributions . Its name follows from the fact that it is derived from the Dirac measure . It is mostly only of theoretical importance and plays an important role in the classification of the infinitely divisible distributions .

definition

The distribution function of ${\displaystyle \delta _{0}}$

A real random variable is called Dirac-distributed to the point , in symbols , if it has the distribution function ${\displaystyle X}$ ${\displaystyle b}$ ${\displaystyle X\sim \delta _{b}}$

${\displaystyle F_{X}(t)={\begin{cases}0&{\text{ falls }}t<b\\1&{\text{ falls }}b\leq t\end{cases}}}$

owns. The distribution of is therefore exactly the Dirac measure at the point , that is, for all measurable quantities applies ${\displaystyle X}$ ${\displaystyle b}$ ${\displaystyle A\subset \mathbb {R} }$

${\displaystyle P(X\in A)={\begin{cases}1&{\text{ falls }}b\in A,\\0&{\text{ sonst.}}\end{cases}}}$

In particular, the random variable almost certainly assumes the value from which the name deterministic distribution can be traced back. ${\displaystyle b}$

properties

Location dimensions

Expectation , mode and median all coincide and are equal to the point ${\displaystyle b}$

Spread

Variance , standard deviation and coefficient of variation coincide and are all the same ${\displaystyle 0}$

symmetry

The Dirac distribution is symmetrical around . ${\displaystyle b}$

Higher moments

The moments are given by

${\displaystyle m_{k}=b^{k}}$

entropy

The entropy of the Dirac distribution is 0.

Accumulators

The cumulative generating function is

${\displaystyle g_{X}(t)=bt}$.

This means that and all other cumulants are equal to 0. ${\displaystyle \kappa _{1}=b}$

Characteristic function

The characteristic function is

${\displaystyle \varphi _{X}(t)=e^{ibt}\,}$

Moment generating function

The moment generating function is

${\displaystyle M_{X}(t)=e^{bt}\,}$

Reproductivity, α-stability and infinite divisibility

The class of Dirac distributions is reproductive , since the sum of Dirac-distributed random variables is again Dirac-distributed, since it is for the convolution

${\displaystyle \delta _{x}*\delta _{y}=\delta _{x+y}}$

applies. Furthermore, Dirac distributions are α-stable distributions with . In some cases, however, Dirac distributions are explicitly excluded from the definition of α stability. In addition, Dirac distributions are infinitely divisible , since . ${\displaystyle \alpha =1}$ ${\displaystyle \delta _{b/n}^{*n}=\delta _{b}}$

Relationship to other distributions

The Dirac distribution usually appears as a degenerate case with poor parameter selection from other distributions. For example, the Bernoulli distribution , the two-point distribution and the binomial distribution are all Dirac distributions if one chooses. Furthermore, the discrete uniform distribution on a point is also a Dirac distribution. ${\displaystyle p=0}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 14.