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→Conditioning on discrete random variables: This was a suitable place to have the expansion in terms of conditional first and second moments |
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When ''X'' takes on countable many values <math>S = \{x_1,x_2,\dots\}</math> with positive probability, i.e., it is a [[discrete random variable]], we can introduce <math>\operatorname{Var}(Y|X=x)</math>, the conditional variance of ''Y'' given that ''X=x'' for any ''x'' from ''S'' as follows:
:<math>\operatorname{Var}(Y|X=x) = \operatorname{E}((Y - \operatorname{E}(Y\mid X=x))^{2}\mid X=x)=\operatorname{E}(Y^2|X=x)-\operatorname{E}(Y|X=x)^2,</math>
where recall that <math>\operatorname{E}(Z\mid X=x)</math> is the [[Conditional_expectation#Conditional_expectation_with_respect_to_a_random_variable|conditional expectation of ''Z'' given that ''X=x'']], which is well-defined for <math>x\in S</math>.
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