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Law Of Total Expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property of conditional expectation, among other names, states that if X is a random variable whose expected value \operatorname(X) is defined, and Y is any random variable on the same probability space, then :\operatorname (X) = \operatorname ( \operatorname ( X \mid Y)), i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. The conditional expected value \operatorname( X \mid Y ), with Y a random variable, is not a simple number; it is a random variable whose value depends on the value of Y. That is, the conditional expected value of X given the ''event'' Y = y is a number and it is a function of y. If we write g(y) for the value of \operatorname ( X \mid Y = y) then the random variable \operatorname( X \mid Y ) is g( Y ) . One special case states that if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then Lebesgue integration, integrate over the measurable subset of interest. This can be expressed as :\nu(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the ''n''-dimensional Euclidean space (corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Of Random Variables
In statistics, the algebra of random variables provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treatment of sums, products, ratios and general functions of random variables, as well as dealing with operations such as finding the probability distributions and the expectations (or expected values), variances and covariances of such combinations. In principle, the elementary algebra of random variables is equivalent to that of conventional non-random (or deterministic) variables. However, the changes occurring on the probability distribution of a random variable obtained after performing algebraic operations are not straightforward. Therefore, the behavior of the different operators of the probability distribution, such as expected values, variances, covariances, and moments, may be different from that observed for the random variable usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Sims
Christopher Albert Sims (born October 21, 1942) is an American econometrician and macroeconomist. He is currently the John J.F. Sherrerd '52 University Professor of Economics at Princeton University. Together with Thomas Sargent, he won the Nobel Memorial Prize in Economic Sciences in 2011. The award cited their "empirical research on cause and effect in the macroeconomy". Biography Sims was born in Washington, D.C., the son of Ruth Bodman (Leiserson), a Democratic politician and daughter of William Morris Leiserson, and Albert Sims, a state department worker. His father was of English and Northern Irish descent, and his mother was of half Estonian Jewish and half English ancestry. His uncle was Yale economist Mark Leiserson. Sims earned his A.B. in mathematics from Harvard University ''magna cum laude'' in 1963 and his PhD in economics from Harvard in 1968 under supervision of Hendrik S. Houthakker. During the 1963–64 academic year, he was a graduate student at the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of the random variable ''Z'' that is formed as the product Z = XY is a ''product distribution''. The product distribution is the PDF of the product of sample values. This is not the same as the product of their PDFs yet the concepts are often ambiguously termed as in "product of Gaussians". Algebra of random variables The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios. Many of these distributions are described in Melvin D. Springer's book from 1979 ''The Algebra of Random Variable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Cumulance
In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of time series. It was introduced by David Brillinger.David Brillinger, "The calculation of cumulants via conditioning", ''Annals of the Institute of Statistical Mathematics'', Vol. 21 (1969), pp. 215–218. It is most transparent when stated in its most general form, for ''joint'' cumulants, rather than for cumulants of a specified order for just one random variable. In general, we have : \kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_i : i\in B \mid Y) : B \in \pi), where * ''κ''(''X''1, ..., ''X''''n'') is the joint cumulant of ''n'' random variables ''X''1, ..., ''X''''n'', and * the sum is over all partitions \pi of the set of indices, and * "''B'' ∈ ;" means ''B'' runs through the whole list of "bloc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Covariance
In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if ''X'', ''Y'', and ''Z'' are random variables on the same probability space, and the covariance of ''X'' and ''Y'' is finite, then :\operatorname(X,Y)=\operatorname(\operatorname(X,Y \mid Z))+\operatorname(\operatorname(X\mid Z),\operatorname(Y\mid Z)). The nomenclature in this article's title parallels the phrase ''law of total variance''. Some writers on probability call this the "conditional covariance formula"Sheldon M. Ross, ''A First Course in Probability'', sixth edition, Prentice Hall, 2002, page 392. or use other names. Note: The conditional expected values E( ''X'' , ''Z'' ) and E( ''Y'' , ''Z'' ) are random variables whose values depend on the value of ''Z''. Note that the conditional expected value of ''X'' given the ''event'' ''Z'' = ''z'' is a function of ''z''. If we write E( ''X'' , ''Z'' = ''z'') = ''g''(''z'') then the rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Variance
The law of total variance is a fundamental result in probability theory that expresses the variance of a random variable in terms of its conditional variances and conditional means given another random variable . Informally, it states that the overall variability of can be split into an “unexplained” component (the average of within-group variances) and an “explained” component (the variance of group means). Formally, if and are random variables on the same probability space, and has finite variance, then: \operatorname(Y) \;=\; \operatorname\bigl operatorname(Y \mid X)\bigr\;+\; \operatorname\!\bigl(\operatorname \mid Xbigr).\! This identity is also known as the variance decomposition formula, the conditional variance formula, the law of iterated variances, or colloquially as Eve’s law, in parallel to the “Adam’s law” naming for the law of total expectation. In actuarial science (particularly in credibility theory), the two terms \operatorname operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Probability
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name. Statement The law of total probability isZwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. page 31. a theorem that states, in its discrete case, if \left\ is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event A :P(A)=\sum_n P(A\cap B_n) or, alternatively, :P(A)=\sum_n P(A\mid B_n)P(B_n), where, for any n, if P(B_n) = 0 , then these terms are simply omitted from the summation since P(A\mid B_n) is finite. The summation can be interpreted as a weighted average, and consequently the marginal probability, P(A), is sometimes called "average probability"; "overall probability" is sometimes used i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Poker
The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information. The fundamental theorem is stated in common language, but its formulation is based on mathematical reasoning. Each decision that is made in poker can be analyzed in terms of the expected value of the payoff of a decision. The correct decision to make in a given situation is the decision that has the largest expected value. If a player could see all of their opponents' cards, they would always be able to calculate the correct decision with mathematical certainty, and the less they deviate from these correct decisions, the better their expected long-term results. This is certainly true heads-up, but Morton's theorem, in which an opponent's correct decision can benefit a player, may apply in multi-way pots. An example Suppose Bob is playing limit Texas hold 'em and is deal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of a set, partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
