Two-point distribution

The two-point distribution is a probability distribution in stochastics . It is a simple discrete probability distribution that is defined on a two-element set . The best known special case is the Bernoulli distribution , which is defined on . ${\displaystyle \{a,b\}}$ ${\displaystyle \{0,1\}}$

definition

A random variable on with is called two-point distribution if ${\displaystyle X}$ ${\displaystyle \{a,b\}}$ ${\displaystyle a<b}$

${\displaystyle P(X=a)=1-p{\text{ und }}P(X=b)=p}$ is.

The distribution function is then

${\displaystyle F_{X}(t)={\begin{cases}0&{\text{ falls }}t<a\\1-p&{\text{ falls }}t\in [a,b)\\1&{\text{ falls }}t\geq b\end{cases}}}$

properties

Be in the following . ${\displaystyle q=1-p}$

Expected value

The expected value of a two-point distributed random variable is

${\displaystyle E(X)=(1-p)\cdot a+p\cdot b=q\cdot a+p\cdot b}$.

Variance and other measures of dispersion

The following applies to the variance

${\displaystyle V(X)=E\left((X-E(X))^{2}\right)=p\cdot q\cdot (b-a)^{2}}$.

Hence the standard deviation

${\displaystyle \sigma _{X}=(b-a){\sqrt {pq}}}$

and the coefficient of variation

${\displaystyle \operatorname {VarK} (X)={\frac {(b-a){\sqrt {pq}}}{qa+pb}}}$.

symmetry

If , the two-point distribution is symmetrical about its expected value. ${\displaystyle p={\tfrac {1}{2}}}$

Crookedness

The skewness of the two-point distribution is

${\displaystyle \operatorname {v} (X)={\frac {1-2p}{\sqrt {pq}}}}$.

Bulge and excess

The excess of the two-point distribution is

${\displaystyle \gamma (X)={\frac {1-6pq}{pq}}}$

and with that is the bulge

${\displaystyle \beta _{2}(X)={\frac {1-3pq}{pq}}}$.

Higher moments

The -th moments result as ${\displaystyle k}$

${\displaystyle \operatorname {E} (X^{k})=qa^{k}+pb^{k}}$.

This can be shown, for example, with the torque generating function.

mode

The mode of two-point distribution is

${\displaystyle x_{D}={\begin{cases}a&{\text{falls }}q>p\\a{\text{ und }}b&{\text{falls }}q=p\\b&{\text{falls }}q<p\end{cases}}}$

Median

The median of the two-point distribution is

${\displaystyle {\tilde {m}}={\begin{cases}a&{\text{falls }}q\geq p\\b&{\text{falls }}q<p\end{cases}}}$

Probability generating function

Are , then is the probability generating function ${\displaystyle a,b\in \mathbb {N} _{0}}$

${\displaystyle m_{X}(t)=qt^{a}+pt^{b}}$.

Moment generating function

The moment-generating function is given for anything as ${\displaystyle a,b\in \mathbb {R} }$

${\displaystyle M_{X}(t)=qe^{at}+pe^{bt}}$.

Characteristic function

The characteristic function is given for anything as ${\displaystyle a,b\in \mathbb {R} }$

${\displaystyle \varphi _{X}(t)=qe^{iat}+pe^{ibt}}$.

Construction of the distribution according to given parameters

If the expected value , standard deviation and skewness are given, a suitable two-point distribution is obtained as follows: ${\displaystyle m}$ ${\displaystyle s}$ ${\displaystyle t}$

${\displaystyle p=(1+t/{\sqrt {4+t^{2}}})/2,}$

${\displaystyle q=1-p,}$

${\displaystyle a=m-s\cdot {\sqrt {q/p}},}$

${\displaystyle b=m+s\cdot {\sqrt {p/q}}.}$

Sums of two-point distributed random variables

The two-point distribution is for non- reproductive . That is, if there are two-point distributions, then there is no longer two-point distributions. The only exception is the degenerate case with (or ). Then it is a Dirac distribution on (or on ), which is accordingly reproductive and even infinitely divisible . ${\displaystyle p\in (0,1)}$ ${\displaystyle X_{1},X_{2}}$ ${\displaystyle X_{1}+X_{2}}$ ${\displaystyle p=1}$ ${\displaystyle q=1}$ ${\displaystyle b}$ ${\displaystyle a}$

Relationship to other distributions

Relationship to the Bernoulli distribution

A two-point distribution on is a Bernoulli distribution . ${\displaystyle \{0,1\}}$

Relationship to the Rademacher distribution

The Rademacher distribution is a two-point distribution with . ${\displaystyle a=-1,b=1,p=q={\frac {1}{2}}}$

literature

• Thomas Mack: Actuarial Science. 2nd Edition. Verlag Versicherungswirtschaft, 2002, ISBN 388487957X .