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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
This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy. The definition of E ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid {\mathcal {H}})} may resemble that of E ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid H)} for an event H {\displaystyle H} but these ...
The tower rule may refer to one of two rules in mathematics: Law of total expectation , in probability and stochastic theory a rule governing the degree of a field extension of a field extension in field theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
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In probability, a generic property is an event that occurs almost surely, meaning that it occurs with probability 1. For example, the law of large numbers states that the sample mean converges almost surely to the population mean. This is the definition in the measure theory case specialized to a probability space.
Integral geometry sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric objects derived from random points, and can in part be viewed as a ...
The term "characteristic function" has an unrelated meaning in classic probability theory. For this reason, traditional probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function [ a ] to describe the function that ...