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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 ...
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
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
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 ...
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 .
The indicator function of A is the Iverson bracket of the property of belonging to A; that is, 1 A ( x ) = [ x ∈ A ] . {\displaystyle \ \mathbf {1} _{A}(x)=\left[\ x\in A\ \right]~.} For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers .
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance.Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements?