Law of Total Variance - Proof

Proof

The law of total variance can be proved using the law of total expectation. First,

from the definition of variance. Then we apply the law of total expectation to each term by conditioning on the random variable X:

Now we rewrite the conditional second moment of Y in terms of its variance and first moment:

Since the expectation of a sum is the sum of expectations, the terms can now be regrouped:

Finally, we recognize the terms in parentheses as the variance of the conditional expectation E: