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Removed underscore X from the main formulate that was introduced by EtudiantEco @ 02:26, 3 February 2015 (similar to as in other article). Again: The underscore X notation is not more precise, but is rather confusing and unnecessary. |
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The [[law of total variance]] says
:<math>\operatorname{Var}(Y) = \operatorname{E}
where, for example, <math>\operatorname{Var}(Y|X)</math> is understood to mean that the value ''x'' at which the conditional variance would be evaluated is allowed to be a [[random variable]], ''X''. In this "law", the inner expectation or variance is taken with respect to ''Y'' conditional on ''X'', while the outer expectation or variance is taken with respect to ''X''. This expression represents the overall variance of ''Y'' as the sum of two components, involving a prediction of ''Y'' based on ''X''. Specifically, let the predictor be the least-mean-squares prediction based on ''X'', which is the [[conditional expectation]] of ''Y'' given ''X''. Then the two components are:
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