Conditional probability distribution: Difference between revisions

Revision as of 02:50, 23 March 2010

Revision as of 10:29, 23 March 2010

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{{unreferenced|date=March 2009}}

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''). Fromusing the definition of [[conditional probability]], this is defined as:

:<math>Pp_Y(Y = y \mid X = x) = \frac{P(XY =x\ y \capmid Y=y)}{P(X = x)} = \frac{P(X = x\ \midcap Y = y) P(Y = y)}{P(X = x)}.</math>

The relation with the probability distribution of ''X'' given ''Y'' is:

Similarly for [[continuous random variable]]s, the conditional [[probability density function]] can be written as ''f''_''Y''(''y'' | ''X=x'') and this is

:<math>P(Y=y \mid X=x) P(X=x) = P(X=x\ \cap Y=y) = P(X=x \mid Y=y)P(Y=y).</math>

Similarly for [[continuous random variable]]s, the conditional [[probability density function]] can be written as ''f''_''Y''(''y'' | ''X=x'') and this is

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