Probability distributions

Probability - (. approximate(limit) :sp

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Probability Distributions Probability distributions have a surprising number inter-connections  (mathematically speaking). A dashed line in the chart below indicates an  approximate (limit) relationship between two distribution families.  A solid line indicates an exact relationship: special case, sum, or transformation. Click on a distribution  for the parameterization of that distribution.  Click on an  arrow  for details on the relationship represented by the arrow. See also a similar diagram for  conjugate prior relationships .
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   The chart above is adapted from the chart originally published by Lawrence  Leemis in 1986 (Relationships Among Common Univariate Distributions,  American Statistician 40:143-146.)  Leemis published a larger chart in 2008 which is available  online
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Parameterizations The precise relationships between distributions depend on parameterization. The  relationships detailed below depend on the following parameterizations for the  PDFs.   Distribution Base name     Parameters beta beta   shape1 shape2 binomial binom size prob Cauchy cauchy location scale chi-squared chisq df exponential exp   rate F f df1 df2 gamma  gamma shape rate geometric geom p Hyper-geometric hyper m n k log-normal lnorm meanlog sdlog logistic logis location scale negative binomial nbinom size prob normal norm mean sd Poisson pois lambda Student t t df uniform  unif min max Weibull weibull shape scale Note that the exponential is parameterized in terms of the rate, the reciprocal of the  mean. The gamma can be parameterized by its shape and either the rate or the scale.  The rate is the default argument by position, but you can specify the scale by name. The 
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hyper-geometric distribution gives the probability of various numbers of red balls when k  balls are taken from an urn containing m red balls and n blue balls. Note that another  popular convention uses the number of red balls and the total number of balls m+n. Note  that the parameters for the log-normal are the mean and standard deviation of the log of  the distribution, not the mean and standard deviation of the distribution itself.   Also, let C(n, k) denote the  binomial coefficient  (n, k) and     B(a, b) =  (a)  (b) /  (a + b). Γ Γ Γ Distributions and Parameterizations: Geometric : f(x) = p (1-p) x  for non-negative integers x. Discrete uniform : f(x) = 1/n for x = 1, 2, . .., n. Negative binomial
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.

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Probability - (. approximate(limit) :sp

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