Conditional probability distribution: Difference between revisions

Revision as of 21:53, 13 February 2025

Revision as of 21:56, 13 February 2025

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==Properties==

Seen as a function of <math>y</math> for given <math>x</math>, <math>P(Y=y|X=x)</math> is a probability mass function and so the sum over all <math>y</math> (or integral if it is a continuousconditional probability density) is 1. Seen as a function of <math>x</math> for given <math>y</math>, it is a [[likelihood function]], so that the sum (or integral) over all <math>x</math> need not be 1.

Additionally, a marginal of a joint distribution can be expressed as the expectation of the corresponding conditional distribution. For instance, <math> p_X(x) = E_{Y}[p_{X|Y}(x \ |\ Y)] </math>.