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depends on the particular value y; it is a function . That same function evaluated at the random variable Y is the conditional expectation .

In particular, if is a discrete random variable assuming possible values with corresponding probabilities , then in the formula for total variance, the first term on the right-hand side becomes

where . Similarly, the second term on the right-hand side becomes

where and . Thus the total variance is given by

A similar formula is applied in analysis of variance,

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