Search results
Results From The WOW.Com Content Network
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 many purposes, it is only necessary to know that an expansion for in terms of iterated commutators of and exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, [2] where the precise coefficients play no role in the argument.)
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:
In probability theory, the law of total variance [1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if and are random variables on the same probability space, and the variance of is finite, then
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 ...
His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated logarithm for empirical characteristic functions. In the intervening years, his work has touched on the areas of probability , ergodic theory , and harmonic analysis .
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:
Language links are at the top of the page across from the title.