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:

Read more about this topic:  Law Of Total Variance

Famous quotes containing the word proof:

    The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.
    Charles Baudelaire (1821–1867)

    The proof of a poet is that his country absorbs him as affectionately as he has absorbed it.
    Walt Whitman (1819–1892)

    a meek humble Man of modest sense,
    Who preaching peace does practice continence;
    Whose pious life’s a proof he does believe,
    Mysterious truths, which no Man can conceive.
    John Wilmot, 2d Earl Of Rochester (1647–1680)