Editing Bayes' theorem (section)

===Random variables===
[[File:Bayes continuous diagram.svg|thumb|Bayes' theorem applied to an event space generated by continuous random variables ''X'' and ''Y'' with known probability distributions. There exists an instance of Bayes' theorem for each point in the [[Domain of a function|domain]]. In practice, these instances might be parametrized by writing the specified probability densities as a [[Function (Mathematics)|function]] of ''x'' and ''y''.]]
Consider a [[sample space]] Ω generated by two [[random variables]] ''X'' and ''Y'' with known probability distributions. In principle, Bayes' theorem applies to the events ''A''&nbsp;=&nbsp;{''X''&nbsp;=&nbsp;''x''} and ''B''&nbsp;=&nbsp;{''Y''&nbsp;=&nbsp;''y''}.

:<math>P( X{=}x  | Y {=} y) = \frac{P(Y{=}y | X{=}x) P(X{=}x)}{P(Y{=}y)}</math>

Terms become 0 at points where either variable has finite [[probability density function|probability density]]. To remain useful, Bayes' theorem can be formulated in terms of the relevant densities (see [[#Derivation|Derivation]]).

====Simple form====
If ''X'' is continuous and ''Y'' is discrete,

:<math>f_{X | Y{=}y}(x) = \frac{P(Y{=}y| X{=}x) f_X(x)}{P(Y{=}y)}</math>

where each <math>f</math> is a density function.

If ''X'' is discrete and ''Y'' is continuous,

:<math> P(X{=}x| Y{=}y) = \frac{f_{Y | X{=}x}(y) P(X{=}x)}{f_Y(y)}.</math>

If both ''X'' and ''Y'' are continuous,

:<math> f_{X| Y{=}y}(x) = \frac{f_{Y | X{=}x}(y) f_X(x)}{f_Y(y)}.</math>

====Extended form====
[[File:Continuous event space specification.svg|thumb|A way to conceptualize event spaces generated by continuous random variables X and Y]]

A continuous event space is often conceptualized in terms of the numerator terms. It is then useful to eliminate the denominator using the [[law of total probability]]. For ''f<sub>Y</sub>''(''y''), this becomes an integral:

:<math> f_Y(y) = \int_{-\infty}^\infty f_{Y| X = \xi}(y) f_X(\xi)\,d\xi .</math>