Distribution of a random variable

This article deals with the transfer of probability measures to other event spaces by means of random variables. For the map that assigns a probability to sets, see probability measure .

The distribution of a random variable is a term from probability theory , a branch of mathematics. The distribution of a random variable makes it possible to extract information from a stochastic model that is “too large” and to assign meaningful probabilities to it again. An example of this is a lottery drawing: When modeling, the probabilities for each individual number combination are first defined. In general, however, one is not interested in the probability of drawing exactly a certain number sequence, but in how great the probability is for "n correct". To do this, a random variable is defined which extracts the information “number of correct persons”. The distribution of these random variables then indicates the probability that you have drawn “n correct”.

definition

Given a random variable from the probability space into the event space . Then it's called through ${\displaystyle X}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle (\Omega ^{*},\Sigma ^{*})}$

${\displaystyle P_{X}(A^{*}):=P(X^{-1}(A^{*})){\text{ für alle }}A^{*}\in \Sigma ^{*}}$

defined mapping the distribution of the random variables below . It defines a probability measure on . Here the archetype of below , i.e. the event . Sometimes there is also written for. ${\displaystyle P_{X}\colon \Sigma ^{*}\to [0,1]}$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle (\Omega ^{*},\Sigma ^{*})}$ ${\displaystyle X^{-1}(A^{*})}$ ${\displaystyle A^{*}}$ ${\displaystyle X}$ ${\displaystyle \{\omega \in \Omega \mid X(\omega )\in A^{*}\}\in \Sigma }$ ${\displaystyle P_{X}}$ ${\displaystyle P\circ X^{-1}}$

Examples

We consider a three-time coin toss as a model, modeled by the probability space with result set ${\displaystyle (\Omega ,\Sigma ,P)}$

${\displaystyle \Omega :=\{0,1\}^{3}}$,

Event system

${\displaystyle \Sigma :={\mathcal {P}}(\Omega )}$

and as a measure of probability, the equal distribution, since the coin is assumed to be fair and the tosses take place independently of one another , i.e.

${\displaystyle P(A)={\frac {\#A}{\#\Omega }}={\frac {\#A}{8}}}$.

The second event space is now defined as

${\displaystyle (\{0,1,2,3\};{\mathcal {P}}(\{0,1,2,3\}))}$,

the random variable counts the successes, so ${\displaystyle X(\omega )=X(\{(\omega _{1},\omega _{2},\omega _{3})\})=\omega _{1}+\omega _{2}+\omega _{3}}$

In order to determine the distribution of these random variables, it is sufficient to go through a generator, i.e. here the individual elementary events. All other probabilities are then obtained by adding the probabilities of the (disjoint) producers. It is then

${\displaystyle P_{X}(\{0\})=P(X^{-1}(\{0\}))=P(\{(0,0,0)\})={\tfrac {1}{8}}}$

${\displaystyle P_{X}(\{1\})=P(X^{-1}(\{1\}))=P(\{(1,0,0),(0,1,0),(0,0,1)\})={\tfrac {3}{8}}}$

${\displaystyle P_{X}(\{2\})=P(X^{-1}(\{2\}))=P(\{(1,1,0),(0,1,1),(1,0,1)\})={\tfrac {3}{8}}}$

${\displaystyle P_{X}(\{3\})=P(X^{-1}(\{3\}))=P(\{(1,1,1)\})={\tfrac {1}{8}}}$.

This is then the distribution of the random variables and a new probability measure on the event space . ${\displaystyle X}$ ${\displaystyle (\{0,1,2,3\};{\mathcal {P}}(\{0,1,2,3\}))}$

Dimension theoretical perspective

From the point of view of measure theory , the distribution of a random variable is an image measure . The probability space corresponds to a special measurement space , the event space is identical to a measurement space and the random variable is a measurable function . Just like the image measure, the distribution of a random variable enables the “relocation” and modification of a probability measure from a measurement space to a measurement space.

Probability measures as distributions

General measures of probability as distributions

If a probability space is given, the probability measure can be represented as the distribution of a random variable in the following way: The event space is duplicated and the identical mapping of after is selected as the random variable . Then the probability measure and the distribution of the random variables agree. This justifies, among other things, the common term “probability distribution” for probability measures. ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle P}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle X}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle P}$ ${\displaystyle P_{X}}$

Real probability measures as distributions on a common probability space

In fact, each probability measure on the event space can be represented as a distribution of a random variable on the probability space. Here, the constant uniform distribution on the interval from 0 to 1 is used to describe the fact that each probability measure is clearly defined by its distribution function . If now is a distribution function of , then the quantile function defined by is chosen as the random variable ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle ((0,1),{\mathcal {B}}((0,1)),{\mathcal {U}}_{(0,1)})}$ ${\displaystyle {\mathcal {U}}_{(0,1)}}$ ${\displaystyle F}$ ${\displaystyle P}$ ${\displaystyle F^{-1}}$

${\displaystyle F^{-1}(u)=\inf\{x\in \mathbb {R} \mid F(x)\geq u\}}$,

where is. This random variable now has the probability measure as a distribution. This statement makes it possible, for example, to examine any random variable in the real numbers for stochastic independence , since they can always be understood as random variables in the same probability space. ${\displaystyle u\in (0,1)}$ ${\displaystyle P}$

Probability measures constructed as distributions

Binomial distribution

The binomial distribution can be fundamentally defined as the distribution of a random variable. To do this, you define the simple coin toss of an unfair coin with the probability space , the number of successes, the amount of events and the probability measure . The n-times independent tossing of the coin is then described by the product model . Now we define a random variable of the product model to by ${\displaystyle \Omega =\{0,1\}}$ ${\displaystyle {\mathcal {A}}:={\mathcal {P}}(\{0,1\})}$ ${\displaystyle P(\{1\})=p=1-P(\{0\})}$ ${\displaystyle (\Omega ^{\otimes n},{\mathcal {A}}^{\otimes n},P^{\otimes n})}$ ${\displaystyle \Omega _{2}=\{0,1,\cdots ,n\}}$

${\displaystyle X(\omega )=X((\omega _{1},\dots ,\omega _{n}))=\omega _{1}+\omega _{2}+\dots +\omega _{n}}$,

This is how this random variable models the number of successes in coin flips. The distribution of the random variables is then the binomial distribution, so . ${\displaystyle n}$ ${\displaystyle X\sim \mathrm {Bin} _{n,p}}$

Geometric distribution and negative binomial distribution

Just like the binomial distribution, the geometric distribution and the negative binomial distribution can be derived from a product model of a coin toss as the distribution of a random variable. In this case, the product model is the infinitely often repeated coin toss, i.e. with the same designations as above . The random variable from the product model in the event space defined by ${\displaystyle (\Omega ^{\otimes \mathbb {N} },{\mathcal {A}}^{\otimes \mathbb {N} },P^{\otimes \mathbb {N} })}$ ${\displaystyle (\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ))}$

${\displaystyle X(\omega )=\min\{i\in \mathbb {N} \,|\,\omega _{i}=1\}}$

then models the waiting time until the first success and has the geometric distribution as the distribution. If you model the waiting time for the nth success, you get the negative binomial distribution.

Generalizations

There are several special cases of the distribution of a random variable. The common distribution of random variables uses several random variables to define a multivariate distribution on a higher-dimensional space. The marginal distribution, on the other hand, is the distribution of a multivariate distribution under a coordinate mapping, it thus reduces the dimensionality of the probability distribution.

A variation of the distribution of a random variable is the conditional distribution and the regular conditional distribution . Both model additional prior knowledge about the outcome of the random experiment. The conditional distribution is easier to handle and is defined by the conditional probability , but has deficits in dealing with zero sets and is not as general. The regular conditional distribution requires the technical concept of the conditional expected value .

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 .
• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , doi : 10.1007 / 978-3-663-09885-0 .
• 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 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .

Web links

Wiktionary: probability distribution  - explanations of meanings, word origins, synonyms, translations