A problem asks the following. The conditional variance of with respect to is the random variable

where is a decomposition of the sample space. Show that

We can read this as the follows. The variance of is the sum of the expectation of its conditional variances and the variance of the conditional expectations. For example, if , then we ought to see a variance in the conditional expectations (since there is only one condition)

In general, the variance of conditional expectations expands to

and the expectation of conditional variances expands to

Adding the two

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