NationMaster - Encyclopedia: Compound Poisson distribution

In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... A probability distribution describes the values and probabilities that a random event can take place. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum

Via the law of total cumulance it can be shown that the moments of X₁ are the cumulants of Y. 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. ... -1... // Cumulants of probability distributions In probability theory and statistics, the cumulants Îºn of the probability distribution of a random variable X are given by In other words, Îºn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions. The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ...

Compound Poisson processes

A compound Poisson process with rate

λ > 0

and jump size distribution G is a continuous-time stochastic process ${,Y(t) : t geq 0 ,}$ given by This article or section should be merged with Poisson process A compound poisson process with rate and jump size distribution G is a parametrized family of random variables given by where, is a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which... In the mathematics of probability, a stochastic process is a random function. ...

where, ${,N(t) : t geq 0,}$ is a Poisson process with rate

λ

, and ${,D_i : i geq 0,}$ are independent and identically distributed random variables, with distribution function G, which are also independent of ${,N(t) : t geq 0,}.,$ It has been suggested that this article be split into multiple articles. ...

A probability distribution describes the values and probabilities that a random event can take place. ... A probability distribution describes the values and probabilities that a random event can take place. ... A logarithmic scale bar. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... Often confused with the multinomial distribution. ... // In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... Zipfs law, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... The Skellam distribution is the discrete probability distribution of the difference N1 âˆ’ N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values Î¼1 and Î¼2. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Not to be confused with Beta function. ... In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [âˆ’R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... This article is about the mathematics of the chi-square distribution. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... This article does not cite its references or sources. ... The folded normal distribution is a probability distribution related to the normal distribution. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, FrÃ©chet and Weibull families also known as type I, II and III extreme value distributions. ... In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In probability theory and statistics, the half-logistic distribution is a continuous probability distributionâ€”the distribution of the absolute value of a random variable following the logistic distribution. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... The scale-inverse-chi-square distribution arises in Bayesian statistics (spam filtering in particular). ... The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... In probability theory and statistics, the LÃ©vy distribution, named after Paul Pierre LÃ©vy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. ... The Maxwellâ€“Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... There are very few or no other articles that link to this one. ... In probability theory and statistics, the noncentral chi-square or noncentral distribution is a generalization of the chi-square distribution. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... The relativistic Breitâ€“Wigner distribution (after Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function [1]: It is most often used to model resonances (i. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and respectively. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... This article needs cleanup. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891â€“1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory, a LÃ©vy skew alpha-stable distribution or just stable distribution, developed by Paul LÃ©vy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... The variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... A probability distribution describes the values and probabilities that a random event can take place. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors Î±. Clockwise from top left: Î±=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ... In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter Î¼ and a scale parameter Ïƒ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... This article or section is incomplete and may require expansion and/or cleanup. ...

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Compound Poisson processes