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The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
For an illustration, consider the example of a dog show (a selected excerpt of Analysis_of_variance#Example). Let the random variable correspond to the dog weight and correspond to the breed. In this situation, it is reasonable to expect that the breed explains a major portion of the variance in weight since there is a big variance in the ...
Examples: Throwing dice, experiments with decks of cards, random walk, and tossing coins. Classical definition : Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability .
The law of total covariance can be proved using the law of total expectation: First, (,) = [] [] [] from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:
I think the example in the "Iterated expectations with nested conditioning sets" is wrong, as it is not generally valid. The quoted reference in the Lindquist and Sargent book is about Markov processes, in which future states only depend on the current states, not on past states.
Alaska holds the all-time U.S. record. The mercury plummeted to 80 degrees below zero on Jan. 23, 1971, in Prospect Creek, north of Fairbanks.
Donald Trump mocked Canadian Prime Minister Justin Trudeau after his top minister’s surprise resignation following a clash on how to handle the president-elect’s looming tariffs.
The bird flu outbreak has taken concerning turns, with more than 60 human cases confirmed. Experts outlined four signs that the virus is going in the wrong direction.