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(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] |
Probability Theory
Probability theory or probability calculus 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 no ... [...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] |
Discrete Distributions
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution, a random variable that can be counted *Discrete space, a simple example of a topological space *Discrete spline interpolation, the discrete analog of ordinary spline interpolation *Discrete time, non-continuous time, which results in discrete-time samples *Discrete variable In mathematics and statistics, a quantitative variable may be continuous or discrete. If it can take on two real values and all the values between them, the variable is con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Actuarial Association
The International Actuarial Association (IAA) is a worldwide association of local professional actuarial associations. History The IAA is the continuation of the ''Comité Permanent des Congrès d’Actuaires'', established in 1895, as an association of individuals. It was renamed the IAA in 1968. At the 26th International Congress of Actuaries, held in Birmingham on 7–12 June 1998, the General Assembly of the International Actuarial Association restructured itself. The restructure created a single unified framework to ensure unity of direction and efficient coordination with respect to issues of a worldwide nature. The major responsibilities of the IAA are now in the hands of the actuarial associations, which bring together the actuaries in their respective countries and is the link between the actuaries and the actuarial associations worldwide. The IAA is the unique international organization dedicated to the research, education and development of the profession and of actua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Panjer Recursion
The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = \sum_^N X_i\, where both N\, and X_i\, are random variables and of special types. In more general cases the distribution of ''S'' is a compound distribution. The recursion for the special cases considered was introduced in a paper by Harry Panjer ( Distinguished Emeritus Professor, University of Waterloo). It is heavily used in actuarial science (see also systemic risk). Preliminaries We are interested in the compound random variable S = \sum_^N X_i\, where N\, and X_i\, fulfill the following preconditions. Claim size distribution We assume the X_i\, to be i.i.d. and independent of N\,. Furthermore the X_i\, have to be distributed on a lattice h \mathbb_0\, with latticewidth h>0\,. : f_k = P _i = hk\, In actuarial practice, X_i\, is obtained by discretisation of the claim density function (upper, lower...). Claim number distribution The number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. The line may be physical – as set by a Surveying, road surveyor, pictorial as in a diagram of a road or roof, or Pure mathematics, abstract. An application of the mathematical concept is found in the grade (slope), grade or gradient in geography and civil engineering. The ''steepness'', incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: *An "increasing" or "ascending" line goes from left to right and has positive slope: m>0. *A "decreasing" or "descending" line goes from left to right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimator Bias
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In statistics, "bias" is an property of an estimator. Bias is a distinct concept from consistency: consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased (see bias versus consistency for more). All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in unbiased estimation of standard deviation); because a biased estima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Generating Function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a random variable ''X'', and to make available the well-developed theory of power series with non-negative coefficients. Definition Univariate case If ''X'' is a discrete random variable taking values ''x'' in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as G(z) = \operatorname (z^X) = \sum_^ p(x) z^x, where p is the probability mass function of X. Note that the subscripted notations G_X and p_X are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all ... [...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] |
Actuarial Science
Actuarial science is the discipline that applies mathematics, mathematical and statistics, statistical methods to Risk assessment, assess risk in insurance, pension, finance, investment and other industries and professions. Actuary, Actuaries are professionals trained in this discipline. In many countries, actuaries must demonstrate their competence by passing a series of rigorous professional examinations focused in fields such as probability and predictive analysis. Actuarial science includes a number of interrelated subjects, including mathematics, probability theory, statistics, finance, economics, financial accounting and computer science. Historically, actuarial science used deterministic models in the construction of tables and premiums. The science has gone through revolutionary changes since the 1980s due to the proliferation of high speed computers and the union of stochastic actuarial models with modern financial theory. Many universities have undergraduate and gradu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which * the domain is the set of possible outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
