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)