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Indecomposable Distribution
In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: ''Z'' ≠ ''X'' + ''Y''. If it can be so expressed, it is decomposable: ''Z'' = ''X'' + ''Y''. If, further, it can be expressed as the distribution of the sum of two or more independent ''identically'' distributed random variables, then it is divisible: ''Z'' = ''X''1 + ''X''2. Examples Indecomposable * The simplest examples are Bernoulli-distributeds: if ::X = \begin 1 & \text p, \\ 0 & \text 1-p, \end :then the probability distribution of ''X'' is indecomposable. :Proof: Given non-constant distributions ''U'' and ''V,'' so that ''U'' assumes at least two values ''a'', ''b'' and ''V'' assumes two values ''c'', ''d,'' with ''a'' < ''b'' and ''c'' < ''d'', then ''U'' + ''V'' ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory 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 not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.B. Mandelbrot, The Pareto–Lévy Law and the Distribution of Income, International Economic Review 1960 https://www.jstor.org/stable/2525289 Of the four parameters defining the family, most attention has been focused on the stability parameter, \alpha (see panel). Stable distributions have 0 < \alpha \leq 2, with the upper bound corresponding to the , and to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi-squared Distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral chi-squared distribution. The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cochran's Theorem
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance. Statement Let ''U''1, ..., ''U''''N'' be i.i.d. standard normally distributed random variables, and U = _1, ..., U_NT. Let B^,B^,\ldots, B^be symmetric matrices. Define ''r''''i'' to be the rank of B^. Define Q_i=U^T B^U, so that the ''Q''i are quadratic forms. Further assume \sum_i Q_i = U^T U. Cochran's theorem states that the following are equivalent: * r_1+\cdots +r_k=N, * the ''Q''''i'' are independent * each ''Q''''i'' has a chi-squared distribution with ''r''''i'' degrees of freedom. Often it's stated as \sum_i A_i = A, where A is idempotent, and \sum_i r_i = N is replaced by \sum_i r_i = rank(A). But after an orthogonal transform, A = diag(I_M, 0), and so we reduce to the above theorem. Proof Claim: Let X be a standard Gaussian in \R^n, then for any symmetric matric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Divisibility (probability)
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function.Lukacs, E. (1970) ''Characteristic Functions'', Griffin , London. p. 107 More rigorously, the probability distribution ''F'' is infinitely divisible if, for every positive integer ''n'', there exist ''n'' i.i.d. random variables ''X''''n''1, ..., ''X''''nn'' whose sum ''S''''n'' = ''X''''n''1 + … + ''X''''nn'' has the same distribution ''F''. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Binomial Distribution
In probability theory Probability theory 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 o ... and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted r) occurs. For example, we can define rolling a 6 on a die as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success (r=3). In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution. An alternative formulation is to model the number of total trials (instead of the number of failures). In fact, for a specified (non-ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \; * The probability distribution of the number ''Y'' = ''X'' − 1 of failures before the first success, supported on the set \. Which of these is called the geometric distribution is a matter of convention and convenience. These two different geometric distributions should not be confused with each other. Often, the name ''shifted'' geometric distribution is adopted for the former one (distribution of the number ''X''); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly. The geometric distribution gives the probability that the first occurrence of success requires ''k'' independent trials, each with success probability ''p''. If the probability of suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitely Divisible Distribution
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function.Lukacs, E. (1970) ''Characteristic Functions'', Griffin , London. p. 107 More rigorously, the probability distribution ''F'' is infinitely divisible if, for every positive integer ''n'', there exist ''n'' i.i.d. random variables ''X''''n''1, ..., ''X''''nn'' whose sum ''S''''n'' = ''X''''n''1 + … + ''X''''nn'' has the same distribution ''F''. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argumentum A Fortiori
''Argumentum a fortiori'' (literally "argument from the stronger eason) (, ) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more certain than, the first. Usage American usage In ''Garner's Modern American Usage'', Garner says writers sometimes use ''a fortiori'' as an adjective as in "a usage to be resisted". He provides this example: "Clearly, if laws depend so heavily on public acquiescence, the case of conventions is an ''a fortiori'' ead ''even more compelling''one." Jewish usage ''A fortiori'' arguments are regularly used in Jewish law under the name kal va-chomer, literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case. Relation with Ancient Indian Logic In ancient Indian logic (nyaya), the instrument of argumentation known as ''kaimutika'' or ''kaimutya nyaya'' is found to have resem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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