Conditional variance: Difference between revisions

Revision as of 21:41, 3 June 2024

Latest revision as of 10:03, 4 June 2024

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{{main|least-squares}}

Recall that variance is the expected squared deviation between a random variable (say, ''Y'') and its expected value.

The expected value can be thought of as a reasonable prediction of the outcomes of the random experiment (in particular, the expected value is the best constant prediction when predictions are assessed by expected squared prediction error). Thus, one interpretation of expected valuevariance is that it gives the smallest possible expected squared prediction error. If we have the knowledge of another random variable (''X'') that we can use to predict ''Y'', we can potentially use this knowledge to reduce the expected squared error. As it turns out, the best prediction of ''Y'' given ''X'' is the conditional expectation. In particular, for any <math>f: \mathbb{R} \to \mathbb{R}</math> measurable,

:<math>