Univariate probability distribution

The univariate (probability) distributions are the largest and most common class of probability distributions in stochastics . Clearly, the univariate probability distributions are those distributions that can be defined on the real numbers. Higher-dimensional counterparts form the multivariate distributions and the matrix-variant probability distributions .

definition

A probability distribution is called a univariate probability distribution if it is defined on a one-dimensional result space.

In most cases these are the natural numbers (provided with the power set as σ-algebra ) or the real numbers (provided with Borel's σ-algebra ). ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$

Examples

Most of the common probability distributions are univariate, many can be found, for example, in the list of univariate probability distributions . They appear as distributions of real-valued random variables .

Some examples are:

• the Bernoulli distribution , defined on and thus also defined on ${\displaystyle \{0,1\}}$ ${\displaystyle \mathbb {N} }$
• the binomial distribution , defined on and thus also defined on. ${\displaystyle \{0,1,2,\dots ,n\}}$ ${\displaystyle \mathbb {N} }$
• the Poisson distribution defined on the natural numbers
• the exponential distribution defined on the interval . ${\displaystyle [0,+\infty )}$

Demarcation

Caution is advised if a distribution is further determined by certain shape parameters, as is the case with the normal distribution : It has both shape parameters . The presence of two of these parameters has no effect on whether the distribution is univariate or not. Only the dimension of the underlying room (in this example ) is relevant. ${\displaystyle \mu \in \mathbb {R} ,\,\sigma ^{2}>0}$ ${\displaystyle \mathbb {R} }$

General quantities are just as problematic, for example

${\displaystyle M:=\{\operatorname {Kopf} ,\operatorname {Zahl} ,\operatorname {Pferd} \}}$,

since it is not clear here what exactly the dimension of the base space is. It is only after coding (head = 1, number = 2, horse = 3) and embedding, for example, in the natural numbers, that it makes sense to speak of a univariate probability distribution.

Generalizations

Typical generalizations of univariate probability distributions are the multivariate distributions ; they are defined on an n-dimensional base space, usually the . Typical examples are the multinomial distribution and the multi-dimensional normal distribution . ${\displaystyle \mathbb {R} ^{n}}$

The matrix-variable probability distributions are a further generalization ; they appear as the distribution of a matrix-valued random variable. An example is the wishart distribution .

literature

• 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 .