Conditional probability distribution

Given two jointly distributed random variables X and Y, the conditional 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 variables, the conditional probability mass function can be written as P(Y=y|X=x). From Bayes' theorem, this is

$P(Y=y|X=x)={\frac {P(X=x,Y=y)}{P(X=x)}}={\frac {P(X=x|Y=y)P(Y=y)}{P(X=x)}}$

Similarly for continuous random variables, the conditional probability density function can be written as p_Y|X(y|x) and this is

$p_{Y|X}(y|x)={\frac {p_{X,Y}(x,y)}{p_{X}(x)}}={\frac {p_{X|Y}(x|y)p_{Y}(y)}{p_{X}(x)}}$

where p_X,Y(x,y) gives the joint distribution of X and Y, while p_X(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.