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)

Comments

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.

References

[a1]	L.P. Breiman, "Probability" , Addison-Wesley (1968)