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Compound Poisson Distribution
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution. Definition Suppose that :N\sim\operatorname(\lambda), i.e., ''N'' is a random variable whose distribution is a Poisson distribution with expected value λ, and that :X_1, X_2, X_3, \dots are identically distributed random variables that are mutually independent and also independent of ''N''. Then the probability distribution of the sum of N i.i.d. random variables :Y = \sum_^N X_n is a compound Poisson distribution. In the case ''N'' = 0, then this is a sum of 0 terms, so the value of ''Y'' is 0. Hence the conditional distribution of ''Y'' given that ''N'' = 0 is a degenerate distribution. The compound Poisson distribution is obtained by marginalising the j ... [...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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...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 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tweedie Distribution
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models. The Tweedie distributions were named by Bent Jørgensen after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984. Definitions The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship. A random variable ''Y'' is Tweedie distributed ''Twp(μ, σ2)'', if Y \sim \mathrm(\mu, \sigma^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. Definitions Probability density function The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin \lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: #With a shape parameter k and a scale parameter \theta. #With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/ \theta , called a rate parameter. In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which E 'X''= ''kθ'' = ''α''/''β'' is fixed and greater than zero, and E n(''X'')= ''ψ''(''k'') + ln(''θ'') = ''ψ''(''α'') − ln(''β'') is fixed (''ψ'' is the digamma function). Definitions The parameterization with ''k'' and ''θ'' appears to be more common in econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Operational Research Society
The ''Journal of the Operational Research Society'' is a peer-reviewed academic journal covering operations research. It is an official journal of The Operational Research Society and has been in existence since 1950. It publishes full length case-oriented papers, full length theoretical papers, technical notes, discussions (viewpoints) and book reviews. History The journal began as ''Operational Research Quarterly'' in 1950. At that time it was published by the Operational Research Club (Great Britain). It was published four times a year until 1978 (from 1953–1969 under the title ''OR'') when it became a monthly publication and the name was changed to ''Journal of the Operational Research Society''. Abstracting and indexing The journal is abstracted and indexed by ABI/INFORM, Compendex, Current Contents/Engineering, Computing & Technology, Current Contents/Social & Behavioural Sciences, Inspec, Science Citation Index, Social Sciences Citation Index, Scopus, and Zentralblatt M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulk Queue
In queueing theory, a discipline within the mathematical theory of probability, a bulk queue (sometimes batch queue) is a general queueing model where jobs arrive in and/or are served in groups of random size. Batch arrivals have been used to describe large deliveries and batch services to model a hospital out-patient department holding a clinic once a week, a transport link with fixed capacity and an elevator. Networks of such queues are known to have a product form stationary distribution under certain conditions. Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion. Kendall's notation In Kendall's notation for single queueing nodes, the random variable denoting bulk arrivals or service is denoted with a superscript, for example M''X''/M''Y''/1 denotes an M/M/1 queue where the arrivals are in batches determined by the random variable ''X'' and the services in bulk determined by the random variable ''Y''. In a similar way, the GI/G/1 qu ... [...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 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luria–Delbrück Experiment
The Luria–Delbrück experiment (1943) (also called the Fluctuation Test) demonstrated that in bacteria, genetic mutations arise in the absence of selective pressure rather than being a response to it. Thus, it concluded Darwin's theory of natural selection acting on random mutations applies to bacteria as well as to more complex organisms. Max Delbrück and Salvador Luria won the 1969 Nobel Prize in Physiology or Medicine in part for this work. History By the 1940s the ideas of inheritance and mutation were generally accepted, though the role of DNA as the hereditary material had not yet been established. It was thought that bacteria were somehow different and could develop heritable genetic mutations depending on the circumstances they found themselves: in short, was the mutation in bacteria pre-adaptive (pre-existent) or post-adaptive (directed adaption)?Luria SE (1984) A slot machine, a broken test tube: An autobiography. Harper & Row In their experiment, Luria and Delbrüc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Poisson Distribution
In probability theory and statistics, the geometric Poisson distribution (also called the Pólya–Aeppli distribution) is used for describing objects that come in clusters, where the number of clusters follows a Poisson distribution and the number of objects within a cluster follows a geometric distribution. It is a particular case of the compound Poisson distribution. The probability mass function of a random variable ''N'' distributed according to the geometric Poisson distribution \mathcal(\lambda,\theta) is given by : f_N(n) = \mathrm(N=n)= \begin \sum_^n e^\frac(1-\theta)^\theta^k\binom, & n>0 \\ e^, & n=0 \end where ''λ'' is the parameter of the underlying Poisson distribution and θ is the parameter of the geometric distribution. The distribution was described by George Pólya in 1930. Pólya credited his student Alfred Aeppli's 1924 dissertation as the original source. It was called the geometric Poisson distribution by Sherbrooke in 1968, who gave probability tables ... [...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] |
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