# Conditional distribution

A function of an elementary event and a Borel set, which for each fixed elementary event is a probability distribution and for each fixed Borel set is a conditional probability.

Let $( \Omega , {\mathcal A} , {\mathsf P} )$ be a probability space, $\mathfrak B$ the $\sigma$- algebra of Borel sets on the line, $X$ a random variable defined on $( \Omega , {\mathcal A} )$ and $\mathfrak F$ a sub- $\sigma$- algebra of ${\mathcal A}$. A function $Q ( \omega , B )$ defined on $\Omega \times \mathfrak B$ is called a (regular) conditional distribution of the random variable $X$ with respect to the $\sigma$- algebra $\mathfrak F$ if:

a) for fixed $B \in \mathfrak B$ the function $Q ( \omega , B )$ is $\mathfrak F$- measurable;

b) with probability one, for fixed $\omega$ the function $Q ( \omega , B )$ is a probability measure on $\mathfrak B$;

c) for arbitrary $F \in \mathfrak F$,

$$\int\limits _ { F } Q ( \omega , B ) {\mathsf P} ( d \omega ) = {\mathsf P} \{ ( X \in B ) \cap F \} .$$

Similarly one can define the conditional distribution of a random element $\mathfrak J$ with values in an arbitrary measurable space $( \mathfrak X , \mathfrak B )$. If $\mathfrak X$ is a complete separable metric space and $\mathfrak B$ is the $\sigma$- algebra of Borel sets, then the conditional distribution of the random element $\mathfrak J$ relative to any $\sigma$- algebra $\mathfrak F$, $\mathfrak F \subset {\mathcal A}$, exists.

The function $F _ {X} ( x \mid \mathfrak F ) = Q ( \omega , ( - \infty , x ] )$ is called the conditional distribution function of the random variable $X$ with respect to the $\sigma$- algebra $\mathfrak F$.

The conditional distribution (conditional distribution function) of a random variable $X$ with respect to a random variable $Y$ is defined as the conditional distribution (conditional distribution function) of $X$ with respect to the $\sigma$- algebra generated by $Y$.

The conditional distribution function $F _ {X} ( x \mid Y )$ of a random variable $X$ with respect to $Y$ is a Borel function of $Y$; for $Y = y$ its value $F _ {X} ( x \mid Y = y )$ is called the conditional distribution function of $X$ for a fixed value of $Y$. If $Y$ has a density $f _ {Y} ( y)$, then

$$F _ {X} ( x \mid Y = y ) = \ \frac{1}{f _ {Y} ( y) } \frac \partial {\partial y } F _ {X,Y} ( x , y ) ,$$

where $F _ {X,Y} ( x , y )$ is the joint distribution function of $X$ and $Y$.

#### References

 [1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) [2] M. Loève, "Probability theory" , Princeton Univ. Press (1963) [3] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) (Translated from Russian)

Another definition of a conditional distribution is as a function $f ( \omega , B )$ of a regular event and a Borel set such that, for fixed $\omega$, $f ( \omega , \cdot )$ is a probability measure and, for fixed $B$, $f ( \cdot , B )$ is a measurable function.