Marginal_distribution

Marginal distribution

Please help improve this article or section by expanding it. Further information might be found on the talk page or at requests for expansion. (April 2008)

In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y.

For discrete random variables, the marginal probability mass function can be written as Pr(X = x). This is

$\Pr(X=x) = \sum_{y} \Pr(X=x,Y=y) = \sum_{y} \Pr(X=x|Y=y) \Pr(Y=y),$

where Pr(X = x,Y = y) is the joint distribution of X and Y, while Pr(X = x|Y = y) is the conditional distribution of X given Y.

Similarly for continuous random variables, the marginal probability density function can be written as p_X(x). This is

$p_{X}(x) = \int_y p_{X,Y}(x,y) \, dy = \int_y p_{X|Y}(x|y) \, p_Y(y) \, dy$

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