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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] |
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] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The Pearson product-moment correlation coefficient, correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables. A distinction must be made between (1) the covariance of ... [...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] |
Conditional Expected Value
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 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 ''A'' = 0 otherwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 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] |
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] |
Covariance And Correlation
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways. If ''X'' and ''Y'' are two random variables, with means (expected values) ''μX'' and ''μY'' and standard deviations ''σX'' and ''σY'', respectively, then their covariance and correlation are as follows: ; covariance :\text_ = \sigma_ = E X-\mu_X)\,(Y-\mu_Y)/math> ; correlation :\text_ = \rho_ = E X-\mu_X)\,(Y-\mu_Y)(\sigma_X \sigma_Y)\,, so that \rho_ = \sigma_ / (\sigma_X \sigma_Y) where ''E'' is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If ''Y'' always takes on the same values as ''X'', we have the covariance of a variable with itself (i.e. \sigma_), which is called the variance and is more commonly denoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles Containing Proofs
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article(s) may also refer to: Government and law * Elements of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries; called articles of incorporation in the US * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of impeachment, a formal document and charge used for impeachment in the United States * Article of manufacture, in the United States patent law, a category of things that may be patented * Articles of organization, for limited liability organizations, a US equivalent of articles of association Other uses * Article element , in HTML * "Articles", a song ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
