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Extended Negative Binomial Distribution
In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution. It is a truncated version of the negative binomial distribution for which estimation methods have been studied.Shah S.M. (1971) "The displaced negative binomial distribution", ''Bulletin of the Calcutta Statistical Association'', 20, 143–152 In the context of actuarial science, the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt when they characterized all distributions for which the extended Panjer recursion works. For the case , the distribution was already discussed by Willmot and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber. Probability mass function For a natural number and real parameters , with and , the probability mass function of the ExtNegBin(, , ) distribution is given by : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...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] |
Truncated Distribution
In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information. Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as censoring, as opposed ... [...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] |
Logarithmic Distribution
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = p + \frac + \frac + \cdots. From this we obtain the identity :\sum_^ \frac \; \frac = 1. This leads directly to the probability mass function of a Log(''p'')-distributed random variable: : f(k) = \frac \; \frac for ''k'' ≥ 1, and where 0 < ''p'' < 1. Because of the identity above, the distribution is properly normalized. The is : where ''B'' is the |
Probability Mass Function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. Formal definition Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function p: \R \to ,1/math> defined by for -\inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...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] |
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