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Probability Distributions and Generating Functions
2012The p-dimensional random variable \(X = {[{\textbf{X} }_{1},\ldots,{X}_{p}]}^{\mbox{ T}}\) has a normal distribution, denoted \({\mbox{ N}}_{p}(\boldsymbol\mu,\boldsymbol\Sigma )\), with mean \(\boldsymbol\mu = {[{\mu }_{1},\ldots,{\mu }_{p}]}^{\mbox{ T}}\) and p ×p variance–covariance matrix \(\Sigma \) if its density is of the form $$p(\textbf{x}
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Integral Probability Metrics and Their Generating Classes of Functions
Advances in Applied Probability, 1997We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly
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Algorithms for generating random variables with a rational probability-generating function
International Journal of Computer Mathematics, 2014Two algorithms for generating random variables with a rational probability-generating function are presented. One of them implements the recently developed general range reduction method, and the other is an extension of the alias method designed for generating discrete finite-valued random variables to the case where the generated random variable is ...
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Polynomial bounds for probability generating functions
Journal of Applied Probability, 1975The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k ...
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Generating Functions in Elementary Probability Theory
The American Statistician, 1972For many years certain authors of textbooks in probability and statistics have used the term "moment generating function" without first defining "generating function." As in Wilks (1, p. 114) the usual definition of "moment generating function" of a random variable X (with moments of all orders) is E(e8X) where "E" stands for "expected value." Also as ...
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On The Left Tail-End Probabilities and the Probability Generating Function
Journal of the Nigerian Association of Mathematical Physics, 2013In this paper, another tail-end probability function is proposed using the left tail-end probabilities, p( ≤ i ) = Πṙ The resulting function, πx(t), is continuous and converges uniformly within the unit circle, | t | < 1. A clear functional link is established between πx(t) and two other well known versions of the probability generating function.
Igabari, JN, Nduka, EC
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On the Recognition and Structure of Probability Generating Functions
1997If $$ M\left( s \right) = 1 - {e^{ - \pi (s)}} $$ is a probability generating function, the coefficients π j in the MacLaurin expansion π(s) comprise a harmonic renewal sequence. A simple sufficient condition is given which ensures that a non-negative sequence is harmonic renewal. This condition covers the case of the limiting conditional law of
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Jensen-information generating function and its connections to some well-known information measures
Statistics and Probability Letters, 2021Omid Kharazmi, N Balakrishnan
exaly