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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 ...
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The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
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:
Absorption is a valid argument form and rule of inference of propositional logic. [1] [2] The rule states that if implies , then implies and .The rule makes it possible to introduce conjunctions to proofs.
The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S n, scaled by n −1, converge to zero, respectively in probability and almost surely:
The earliest stopping time for reaching crossing point a, := {: =}, is an almost surely bounded stopping time.Then we can apply the strong Markov property to deduce that a relative path subsequent to , given by := (+), is also simple Brownian motion independent of .