|
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] |
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. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Binomial Distribution
In probability theory 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-random) number of successes (r), the number of failures (n - r) are random because the total trials (n) are random. For example, we could use the negative binomial distribution to model the number of days n (random) a certain machine works (specified by r) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 resulting ... [...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). 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 empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of ''k'': for example the logarithmic distributio ... [...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. 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).Shmueli G., Minka T., ... [...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. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the tuple (X, \mathcal A) is called a 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-algebra would be: \mathcal A_1 = \. Then \left(X, \mathcal A_1\right) is a measurable space. Another possible \sigma-algebra would be the power set on X: \mathcal A_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal A_2\right). Common measurable spaces If X is finite or countably infinite, the \sigma-algebra is most often the power set on X, so \mathcal A = \mathcal P(X). This leads to the measurable space (X, \mathcal P(X)). If X is a topological space In mathematics, a topological space is, rou ... [...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 mea ... [...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 obtai ... [...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 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 In probability theory, an intensity measure is a measure that is derived from a random measure. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levy Processes
Levy, Lévy or Levies may refer to: People * Levy (surname), people with the surname Levy or Lévy * Levy Adcock (born 1988), American football player * Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker * Levy Fidelix (1951–2021), Brazilian conservative politician, businessman and journalist * Levy Gerzberg (born 1945), Israeli-American entrepreneur, inventor, and business person * Levy Li (born 1987), Miss Malaysia Universe 2008–2009 * Levy Mashiane (born 1996), South African footballer * Levy Matebo Omari (born 1989), Kenyan long-distance runner * Levy Mayer (1858–1922), American lawyer * Levy Middlebrooks (born 1966), American basketball player * Levy Mokgothu, South African footballer * Levy Mwanawasa (1948–2008), President of Zambia from 2002 * Levy Nzoungou (born 1998), Congolese-French rugby player, playing in England * Levy Rozman (born 1995), American chess IM, coach, and content creator * Levy Sekgapane (born 1990), South A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Equations (Markov Jump Process)
In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evolution of the process's distribution. This article, as opposed to the article titled Kolmogorov equations, focuses on the scenario where we have a continuous-time Markov chain (so the state space \Omega is countable). In this case, we can treat the Kolmogorov equations as a way to describe the probability P(x,s;y,t), where x, y \in \Omega (the state space) and t > s, t,s\in\mathbb R_ are the final and initial times, respectively. The equations For the case of a countable state space we put i,j in place of x,y. The Kolmogorov forward equations read : \frac(s;t) = \sum_k P_(s;t) A_(t) , where A(t) is the transition rate matrix (also known as the generator matrix), while the Kolmogorov backward equations are : \frac(s;t) = -\sum_k A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
