Conditional density

The density of a conditional distribution. Let $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ be a probability space, let $ \mathfrak B $ be the $ \sigma $- algebra of Borel sets on the line, let $ \mathfrak F $ be a sub- $ \sigma $- algebra of $ {\mathcal A} $, let

$$ Q ( \omega , B ) = \ {\mathsf P} \{ X \in B \mid \mathfrak F \} ,\ \ \omega \in \Omega ,\ \ B \in \mathfrak B , $$

be the conditional distribution of $ X $ with respect to $ \mathfrak F $, and let

$$ F _ {X} ( x \mid \mathfrak F ) \ = Q ( \omega , ( - \infty , x ) ) $$

be the conditional distribution function of $ X $ with respect to $ \mathfrak F $. If

$$ F _ {X} ( x \mid \mathfrak F ) = \ \int\limits _ {- \infty } ^ { x } f _ {X} ( t \mid \mathfrak F ) d t , $$

then $ f _ {X} ( x \mid \mathfrak F ) $ is called the conditional density of the distribution of $ X $ with respect to the $ \sigma $- algebra $ \mathfrak F $.

If $ X $ and $ Y $ are random variables, $ f _ {Y} ( y) $ is the density of the distribution of $ Y $ and $ f _ {X,Y} ( x , y ) $ is the joint density of the distribution of $ X $ and $ Y $, then

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

defines the conditional density of the distribution of the random variable $ X $ for fixed values $ y $ of $ Y $ for which $ f _ {Y} ( y) \neq 0 $.

References

[1]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)