Conditional distribution

In stochastics, the conditional distribution of random variables is a way of changing a multivariate distribution with the help of the marginal distributions so that the newly created distribution takes into account existing knowledge about the values ​​of one or more random variables. Conditional distributions play an important role in Bayesian statistics , for example in defining the posterior probabilities . The conditional distribution is based on the concept of (elementary) conditional probability and therefore has deficits in terms of general validity and in dealing with zero quantities . The much more general regular conditional distribution , which is based on the conditional expected value , does not have these structural problems, but is also far more technical.

definition

Discreet case

Given a two-dimensional random variable on with joint probability function and the marginal distribution with respect and the appropriate marginal probability function . Then means for the random variable with probability function ${\displaystyle Z=(X,Y)}$ ${\displaystyle \mathbb {Z} ^{2}}$ ${\displaystyle f(x,y)}$ ${\displaystyle Y}$ ${\displaystyle f_{Y}(y)}$ ${\displaystyle P(Y=y)>0}$

${\displaystyle f(x|y):={\frac {P(X=x,Y=y)}{P(Y=y)}}={\frac {f(x,y)}{f_{Y}(y)}}}$

the conditional distribution of given ${\displaystyle X}$ ${\displaystyle Y=y}$ , the probability function is also called the conditional probability function. The associated probability measure is usually referred to as. ${\displaystyle P_{X|Y=y}}$

Steady fall

Given a random variable on . The random variable, which as a distribution function is the conditional distribution function ${\displaystyle Z=(X,Y)}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle F(x|y)=\lim _{h\downarrow 0}P(X\leq x|y\leq Y\leq y+h)}$

possesses is called the conditional distribution of given ${\displaystyle X}$ ${\displaystyle Y=y}$ .

If there is a common density of and and the boundary density exists with respect to and is not equal to zero, then the conditional distribution has the conditional density ${\displaystyle f(x,y)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f_{Y}(y)}$ ${\displaystyle Y}$

${\displaystyle f(x|y)={\frac {f(x,y)}{f_{Y}(y)}}}$.

example

As an example, consider a multinomial random variable , that is . It has the probability function ${\displaystyle Z=(X,Y)}$ ${\displaystyle Z\sim M(n,(p,1-p))}$

${\displaystyle f_{Z}(x,y)\;=\;{\begin{cases}{n \choose x,y}\;p^{x}(1-p)^{y}&{\mbox{wenn }}x+y=n\\0&{\mbox{sonst}}\end{cases}}}$,

the marginal probability re is binomially distributed , so is ${\displaystyle X}$

${\displaystyle f_{X}(x)={n \choose x}p^{x}(1-p)^{(n-x)}}$.

The following then results for the conditional probability function

${\displaystyle f(y|x)={\frac {f_{Z}(x,y)}{f_{X}(x)}}={\begin{cases}1&{\text{ falls }}y=n-x\\0&{\text{ sonst }}\end{cases}}}$.

This is not surprising since the two random variables are coupled with each other. The sum of the successes must always result, therefore the result of already determines the result of . So here the conditional probability is deterministic. ${\displaystyle x+y=n}$ ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle Y}$

literature

• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .