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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
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It is unanimously agreed that statistics depends somehow on probability. But, as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since the Tower of Babel.
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
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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 ...