The proposition in probability theory known as the law of total expectation, the law of iterated expectations, Adam's law, the tower rule, the smoothing theorem, among other names, states that if X is an integrable random variable (i.e., a random variable satisfying E( | X | ) < ∞) and Y is any random variable, not necessarily integrable, on the same probability space, then
i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X.
The nomenclature used here parallels the phrase law of total probability. See also law of total variance.
(The conditional expected value E( X | Y ) is a random variable in its own right, whose value depends on the value of Y. Notice that the conditional expected value of X given the event Y = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Y = y) = g(y) then the random variable E( X | Y ) is just g(Y).
Read more about Law Of Total Expectation: Proof in The Discrete Case, Iterated Expectations With Nested Conditioning Sets
Famous quotes containing the words law of, law, total and/or expectation:
“If one mistreats citizens of foreign countries, one infringes upon one’s duty toward one’s own subjects; for thus one exposes them to the law of retribution.”
—Johann Wolfgang Von Goethe (1749–1832)
“Nobody dast blame this man.... For a salesman, there is no rock bottom to the life. He don’t put a bolt to a nut, he don’t tell you the law or give you medicine. He’s a man way out there in the blue, riding on a smile and a shoeshine. And when they start not smiling back—that’s an earthquake. And then you get yourself a couple of spots on your hat, and you’re finished. Nobody dast blame this man. A salesman is got to dream, boy. It comes with the territory.”
—Arthur Miller (b. 1915)
“The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience.”
—Willard Van Orman Quine (b. 1908)
“A youthful mind is seldom totally free from ambition; to curb that, is the first step to contentment, since to diminish expectation is to increase enjoyment.”
—Frances Burney (1752–1840)