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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). 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] |
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] |
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] |
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 they are typically obtained by ''measuring'' or ''counting'', respectively. If it can ta ... [...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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolute ... [...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 estimato ... [...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 in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf :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 ''pX'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its distr ... [...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] |
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. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (discrete)
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a unif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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