Multivariate distribution function

A multivariate distribution function is a real-valued function in stochastics that is used to investigate multivariate probability distributions and the distribution of random vectors . It is the higher-dimensional counterpart of the univariate distribution function and, like this, gains its importance from the fact that, according to the correspondence theorem, the multivariate probability distributions can be clearly characterized by their multivariate distribution function. This allows the investigation of probability distributions with methods of measurement theoryreduce to the more accessible investigation of real-valued functions using methods of multidimensional real analysis .

In addition to the designation as a multivariate distribution function, there is also the n-dimensional distribution function , or distribution function, ${\displaystyle \mathbb {R} ^{n}}$ as a designation or, to better differentiate it from the related concept of the multivariate distribution function, the designation multi-dimensional distribution function in the narrower sense (i. E. S.) .

Notations

The comparison operations for vectors from are to be understood component-wise, that is ${\displaystyle x,y}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle x\leq y}$exactly if for everyone . ${\displaystyle x_{i}\leq y_{i}}$ ${\displaystyle i\in \{1,2,\dots ,n\}}$

Furthermore, be for ${\displaystyle x\in \mathbb {R} ^{n}}$

${\displaystyle (-\infty ,x]:=\{y\in \mathbb {R} ^{n}\,|\,y\leq x\}}$

or defined via the components

${\displaystyle (-\infty ,x]=(-\infty ,x_{1}]\times (-\infty ,x_{2}]\times \dots \times (-\infty ,x_{n}]}$

definition

With the above notations, the definition of the multivariate distribution function is essentially transferred directly from the univariate distribution function.

Is a multivariate probability distribution , ie a probability measure on , it means the function ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$

${\displaystyle F_{P}\colon \mathbb {R} ^{n}\to [0,1]}$

defined by ${\displaystyle F_{P}(x):=P((-\infty ,x])}$

the multivariate distribution function of . ${\displaystyle P}$

Is a -dimensional random vector , then is called ${\displaystyle X}$ ${\displaystyle n}$

${\displaystyle F_{X}\colon \mathbb {R} ^{n}\to [0,1]}$

defined by

${\displaystyle F_{X}(x):=P(X\leq x)}$

the multivariate distribution function of . The multivariate distribution function of a random vector is thus exactly the multivariate distribution function of the distribution of the random vector. ${\displaystyle X}$

The component-wise definition as

${\displaystyle F_{X}(x_{1},x_{2},\dots ,x_{n}):=P(X_{1}\leq x_{1},X_{2}\leq x_{2},\dots ,X_{n}\leq x_{n})}$,

where is. Thus the multivariate distribution function of a random vector is exactly the common distribution function of the components. ${\displaystyle X=(X_{1},X_{2},\dots ,X_{n})^{T}}$

properties

The following applies to every distribution function : ${\displaystyle F=F_{P}}$

• It is continuous on the right in each of its variables
• It is square-monotonous , which means that it always follows. For notation see difference operator . ${\displaystyle a\leq b}$ ${\displaystyle \Delta _{a}^{b}F\geq 0}$ ${\displaystyle \Delta _{a}^{b}}$
• The following applies to the limit values

${\displaystyle \lim _{\min _{i}x_{i}\to \infty }F(x)=1}$ and

${\displaystyle \lim _{\min _{i}x_{i}\to -\infty }F(x)=0}$

Conversely, according to the multivariate version of the correspondence theorem , every function that fulfills the above conditions is a distribution function of a uniquely determined multivariate probability measure.

literature

• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 107.
2. ↑ Kusolitsch: Measure and probability theory. 2014, pp. 74-75.