Conditional variance: Difference between revisions

Revision as of 06:25, 2 June 2016

Revision as of 01:06, 16 July 2016

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using the [[conditional distribution]] of ''Y'' given ''X'' (this exists in this case, as both here ''X'' and ''Y'' are real-valued).

In particular, letting <math>P_{Y|X}</math> be the (regular) [[conditional distribution]] <math>P_{Y|X}</math> of ''Y'' given ''X'', i.e., <math>P_{Y|X}:\Omegamathcal{B} \times \mathbb{R}\to [0,1]</math> (the intention is that <math>P_{Y|X}(U,x) = P(Y\in U|X=x)</math> almost surely over the support of ''X''), we can define

<math> \operatorname{Var}(Y|X=x) = \int (y- \int y' P_{Y|X}(dy'|x))^2 P_{Y|X}(dy|x). </math>