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Mixed Binomial Process
A mixed binomial process is a special point process in probability theory. They naturally arise from restrictions of ( mixed) Poisson processes bounded intervals. Definition Let P be a probability distribution and let X_i, X_2, \dots be i.i.d. random variables with distribution P . Let K be a random variable taking a.s. (almost surely) values in \mathbb N= \ . Assume that K, X_1, X_2, \dots are independent and let \delta_x denote the Dirac measure on the point x . Then a random measure \xi is called a mixed binomial process iff it has a representation as : \xi= \sum_^K \delta_ This is equivalent to \xi conditionally on \ being a binomial process based on n and P . Properties Laplace transform Conditional on K=n , a mixed Binomial processe has the Laplace transform : \mathcal L(f)= \left( \int \exp(-f(x))\; P(\mathrm dx)\right)^n for any positive, measurable function f . Restriction to bounded sets For a point process \xi and a bounded meas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Point Process
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. , .Daley, D.J, Vere-Jones, D. (1988). ''An Introduction to the Theory of Point Processes''. Springer, New York. , . Point processes can be used for spatial data analysis,Diggle, P. (2003). ''Statistical Analysis of Spatial Point Patterns'', 2nd edition. Arnold, London. . which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. There are different mathematical interpretations of a point process, such as a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binomial Process
A binomial process is a special point process in probability theory. Definition Let P be a probability distribution and n be a fixed natural number. Let X_1, X_2, \dots, X_n be i.i.d. random variables with distribution P , so X_i \sim P for all i \in \. Then the binomial process based on ''n'' and ''P'' is the random measure : \xi= \sum_^n \delta_, where \delta_=\begin1, &\textX_i\in A,\\ 0, &\text.\end Properties Name The name of a binomial process is derived from the fact that for all measurable sets A the random variable \xi(A) follows a binomial distribution with parameters P(A) and n : : \xi(A) \sim \operatorname(n,P(A)). Laplace-transform The Laplace transform of a binomial process is given by : \mathcal L_(f)= \left \int \exp(-f(x)) \mathrm P(dx) \rightn for all positive measurable functions f . Intensity measure The intensity measure \operatorname\xi of a binomial process \xi is given by : \operatorname\xi =n P. Generalizations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resultin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Poisson Distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Poisson-type Random Measures
Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution. The PT family of distributions is also known as the Katz family of distributions, the Panjer or (a,b,0) class of distributions and may be retrieved through the Conway–Maxwell–Poisson distribution. Throwing stones Let K be a non-negative integer-valued random variable K\in\mathbb_=\mathbb_\cup\) with law \kappa, mean c\in(0,\infty) and when it exists variance \delta^2>0. Let \nu be a probability measure on the measurable space (E,\mathcal). Let \mathbf=\ be a collection of iid random variables (stones) taking values in (E,\mathcal) with law \nu. The random counting measure N on (E,\mathcal) depends on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Measurable Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. History The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in ''Essai philosophique sur les probabilités'' (1814), and the integral form of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Random Measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. Definition Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let E be a separable complete metric space and let \mathcal E be its Borel \sigma -algebra. (The most common example of a separable complete metric space is \R^n ) As a transition kernel A random measure \zeta is a ( a.s.) locally finite transition kernel from a (abstract) probability space (\Omega, \mathcal A, P) to (E, \mathcal E) . Being a transition kernel means that *For any fixed B \in \mathcal \mathcal E , the mapping : \omega \mapsto \zeta(\omega,B) :is measurable from (\Omega, \mathcal A) to (E, \mathcal E) *For every fixed \omega \in \Omega , the mapping : B \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Dirac Measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. Definition A Dirac measure is a measure on a set (with any -algebra of subsets of ) defined for a given and any (measurable) set by :\delta_x (A) = 1_A(x)= \begin 0, & x \not \in A; \\ 1, & x \in A. \end where is the indicator function of . The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome in the sample space . We can also say that the measure is a single atom at ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. The Dirac measures are the extreme points of the convex set of probability measures on . The name is a back-formation from the Dirac delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Independence (probability Theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |