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Given two jointly distributed [[random variable]]s ''X'' and ''Y'', the '''conditional probability distribution''' of ''Y'' given ''X'' (written "''Y'' | ''X''") is the [[probability distribution]] of ''Y'' when ''X'' is known to be a particular value.
For [[discrete random variable]]s, the [[conditional probability]] mass function can be written as ''P''(''Y'' = ''y'' | ''X'' = ''x''). From the definition of [[conditional probability]], this is
:<math>P(Y = y \mid X = x) = \frac{P(X=x\ \cap Y=y)}{P(X=x)}= \frac{P(X = x \mid Y = y) P(Y = y)}{P(X = x)}.</math>
Similarly for [[continuous random variable]]s, the conditional [[probability density function]] can be written as ''p''<sub>''Y''|''X''</sub>(''y'' | ''x'') and this is
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where ''p''<sub>''X'',''Y''</sub>(x, y) gives the [[joint distribution]] of ''X'' and ''Y'', while ''p''<sub>''X''</sub>(''x'') gives the [[marginal distribution]] for ''X''.
The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: [[Borel's paradox]] shows that conditional probability density functions need not be invariant under coordinate transformations.
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== See also ==
*[[Conditioning (probability)]]
*[[Conditional probability]]
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