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2005-12-29T23:10:07Z

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In [[probability theory]], given two jointly distributed [[random variable]]s ''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 variable]]s, the [[marginal probability]] mass function can be written as Pr(''X'' = ''x''). This is

:<math>\Pr(X=x) = \sum_{y} \Pr(X=x,Y=y) = \sum_{y} \Pr(X=x|Y=y) \Pr(Y=y),</math>

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 variable]]s, the marginal [[probability density function]] can be written as ''p''''X''(''x''). This is

:<math>p_{X}(x) = \int_y p_{X,Y}(x,y) \, dy = \int_y p_{X|Y}(x|y) \, p_Y(y) \, dy </math>

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''.

Why the name 'marginal'? One explanation is to imagine the ''p''(''x'',''y'') in a 2D table such as a spreadsheet. The marginals are got by summing the columns (or rows) -- the column sum would then be written in the margin of the table, ie. the column at the side of the table.

== External links ==
* [http://en.wikipedia.org/wiki/Marginal_distribution Wikipedia article on '''Marginal distribution''']

[[Category:Academic Research]]
[[Category:Statistics]]

Marginal distribution - Revision history

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