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Poisson-type Random Measure
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 pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite. The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X into a measurable space by taking the power set of X as the sigma-algebra \Sigma; that is, all subsets of X are measurable sets. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma \to ,+\infty/math> defined by \mu(A) = \begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A. The counting measure on (X,\Sigma) is σ-finite if and only if the space X is countable In mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions. Under a Poisson distribution with the expectation of ''λ'' events in a given interval, the probability of ''k'' events in the same interval is: :\frac . For instance, consider a call center which receives an average of ''λ ='' 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Binomial Distribution
In probability theory and statistics, the negative binomial distribution, also called a Pascal 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/constant/fixed number of successes r occur. For example, we can define rolling a 6 on some dice 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-random) number of successes , the number of failures is random because the number of total trials is random. For example, we could use the negative binomial distribution to model the number of days (random) a certain machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory), experiments, each asking a yes–no question, and each with its own Boolean-valued function, Boolean-valued outcome (probability), outcome: ''success'' (with probability ) or ''failure'' (with probability ). 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., , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . If the sampling is carried out without replacement, the draws ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(a,b,0) Class Of Distributions
In probability theory, a member of the (''a'', ''b'', 0) class of distributions is any distribution of a discrete random variable ''N'' whose values are nonnegative integers whose probability mass function satisfies the recurrence formula : \frac = a + \frac, \qquad k = 1, 2, 3, \dots for some real numbers ''a'' and ''b'', where p_k = P(N = k). The (a,b,0) class of distributions is also known as the Panjer, the Poisson-type or the Katz family of distributions, and may be retrieved through the Conway–Maxwell–Poisson distribution. Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of ''pj'' and applying the recursion to define subsequent values. This can be of use in fitting distributions to empi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway–Maxwell–Poisson Distribution
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case (mathematics), limiting case. Background The CMP distribution was originally proposed by Conway and Maxwell in 1962 as a solution to handling queueing systems with state-dependent service rates. The CMP distribution was introduced into the statistics literature by Boatwright et al. 2003 Boatwright, P., Borle, S. and Kadane, J.B. "A model of the joint distribution of purchase quantity and timing." Journal of the American Statistical Association 98 (2003): 564–572. and Shmueli et al. (2005). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, and volume with a set X of 'points' in the space, but ''regions'' of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region. Definition Consider a set X and a σ-algebra \mathcal F on X. Then the tuple (X, \mathcal F) is called a measurable space. The elements of \mathcal F are called measurable sets within the measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Functional
In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properties of metric spaces. One type of Laplace functional,D. Stoyan, W. S. Kendall, and J. Mecke. ''Stochastic geometry and its applications'', volume 2. Wiley, 1995.D. J. Daley and D. Vere-Jones. ''An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods'', Springer, New York, second edition, 2003. also known as a characteristic functional is defined in relation to a point process, which can be interpreted as random counting measures, and has applications in characterizing and deriving results on point processes.Barrett J. F. The use of characteristic functionals and cumulant generating functionals to discuss the effect of noise in linear systems, J. Sound & Vibration 1964 vol.1, no.3, pp. 229-238 Its definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Random Measure
Let (E, \mathcal A, \mu) be some measure space with \sigma- finite measure \mu. The Poisson random measure with intensity measure \mu is a family of random variables \_ defined on some probability space (\Omega, \mathcal F, \mathrm) such that i) \forall A\in\mathcal,\quad N_A is a Poisson random variable with rate \mu(A). ii) If sets A_1,A_2,\ldots,A_n\in\mathcal don't intersect then the corresponding random variables from i) are mutually independent. iii) \forall\omega\in\Omega\;N_(\omega) is a measure on (E, \mathcal ) Existence If \mu\equiv 0 then N\equiv 0 satisfies the conditions i)–iii). Otherwise, in the case of finite measure \mu, given Z, a Poisson random variable with rate \mu(E), and X_, X_,\ldots, mutually independent random variables with distribution \frac, define N_(\omega) = \sum\limits_^ \delta_(\cdot) where \delta_(A) is a degenerate measure located in c. Then N will be a Poisson random measure. In the case \mu is not finite the measure N can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ... of a binomial process is given by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
