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# Topic: Negative binomial distribution

 APPENDIX I: THE NEGATIVE BINOMIAL DISTRIBUTION We have derived the Poisson Distribution from the Binomial Distribution, and the necessary condition for the Binomial Distribution to hold is that the probability, p, of an event E shall remain constant for all occurrences of its context-events. The parameters of the distribution are the arithmetic mean (m) and the exponent k. Unlike the positive Binomial, k is not necessarily an integer in the Negative Binomial Distribution. www.fao.org /DOCREP/003/X6602E/x6602e0d.htm   (747 words)

 Journal of Nanjing Agricultural University 1996, 19(3): 55~58 The fundamental components of the spatial distributions of Aphis glycines are aggregated distribution of individual populations; the degrees of aggregation increase with the population densities. The distribution patterns were analyzed using the frequency mapping, aggregation degree indices and regression model methods respectively. glycines are all clustered distributions in all densities and the aggregation degrees increase with the population densities. www.k-state.edu /issa/aphids/reporthtml/trans26.htm   (1202 words)

 The Negative Binomial Distribution. Used as an extension of the Poisson used for the study of rare events This version of the negative binomial distribution is a generalization of the Poisson as used to study the distribution of accidents and events at the individual level. One of the problems with the negative binomial distribution is that there does not seem to be a clear meaning which can be easily given to the variance (Arbous AG, Kerrich JE, 1951). Thus, after fitting the empirical distribution to the Negative Binomial distribution, and estimating the variance, as a measure of accident or disease proneness in the population, you are confronted with the question of what this measure means for individuals or groups in the population, or the development of policy or decisions. home.clara.net /sisa/negb1hlp.htm   (560 words)

 Glossary (Part 2) geometric distribution is a special case of the negative binomial distribution where k=1. Distributions with a longer upper tail are said to be positively (right) skewed, while those with a longer lower tail are negatively (left) skewed. The skewness of data is usually measured through a coefficient of skewness which is zero for symmetric distributions such as the normal or uniform distribution, is greater than zero for positively skewed data, and is less than zero for negatively skewed distributions. www.mrs.umn.edu /~sungurea/statlets/usermanual/glossary2.htm   (5170 words)

 Negative binomial distribution - Wikipedia, the free encyclopedia Third, the negative binomial distribution arises as a continuous mixture of Poisson distributions where the mixing distribution of the Poisson rate is a gamma distribution. Suppose Y is a random variable with a binomial distribution with parameters n and p. Thus the negative binomial distribution bears the same relationship to the negative-integer-exponent case of the binomial theorem that the binomial distribution bears to the positive-integer-exponent case. en.wikipedia.org /wiki/Negative_binomial_distribution   (1637 words)

 Probability Distributions (Statistics Toolbox) In its simplest form, the negative binomial distribution models the number of successes before a specified number of failures is reached in an independent series of repeated identical trials. This form of the negative binomial has no interpretation in terms of repeated trials, but, like the Poisson distribution, it is useful in modelling count data. The negative binomial distribution is more general than the Poisson, and is often suitable for count data when the Poisson is not. www-rohan.sdsu.edu /doc/matlab/toolbox/stats/prob_d21.html   (575 words)

 The Negative Binomial Distribution. As the marginal distribution of the Binomial   (Site not responding. Last check: 2007-10-03) The Geometric Distribution is a special case of the negative binomial distribution with the number positive, in the second blue box, being '1' (one). The negative binomial distribution does basically the same thing as the Binomial, except that now we are asking about the probability of a particular sample size, given that we have found 'x' results to be positive (or 'white', or 'car crashes'), whereas we had expected to find 'u' results to be positive. Input is the same as for the binomial, but now the output expresses a change in the number of cases and not, as in the binomial, the number found positive. home.clara.net /sisa/negb2hlp.htm   (350 words)

 Binomial distribution - Wikipedia, the free encyclopedia In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. en.wikipedia.org /wiki/Binomial_distribution   (1134 words)

 Discrete Probability Distribution - NEGATIVE BINOMIAL DISTRIBUTION If the experiment is repeated indefinitely and the trials are independent of each other, then the random variable X, whose value is the number of the trial on which the rth success occurs, has a negative binomial distribution with parameters r and p. A random variable X, having a negative binomial distribution with parameters r and p, is the sum of r independent random variables, each one geometrically distributed with parameter p. In fact, a geometric distribution with parameter p is the same as a negative binomial distribution with parameters n = 1 and p. library.thinkquest.org /10030/6dpdnbd.htm   (311 words)

 Negative Binomial Distribution: Probability Calculator   (Site not responding. Last check: 2007-10-03) The probability distribution of a negative binomial random variable is called a negative binomial distribution. With a negative binomial experiment, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed. With a negative binomial distribution, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed. www.stattrek.com /Tables/NegBinomial.aspx   (1205 words)

 PlanetMath: negative binomial random variable is a negative binomial random variable with parameters "negative binomial random variable" is owned by bgins. This is version 4 of negative binomial random variable, born on 2001-10-26, modified 2004-02-14. planetmath.org /encyclopedia/NegativeBinomialDistribution.html   (55 words)

 Functions and CALL Routines : CDF The CDF function for the binomial distribution returns the probability that an observation from a binomial distribution, with parameters p and n, is less than or equal to m. The CDF function for the F distribution returns the probability that an observation from an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and noncentrality parameter nc, is less than or equal to x. The CDF function for the negative binomial distribution returns the probability that an observation from a negative binomial distribution, with probability of success p and number of successes n, is less than or equal to m. www.asu.edu /sas/sasdoc/sashtml/lgref/z0208980.htm   (1101 words)

 XLSTAT, Statistical software for MS Excel - Statistics and data analysis with MS Excel : Tutorials : How do I create ... Because we want to test the fit between the negative binomial distribution function and the sample, (the Chi-square test requires that there is are least 5 data in a class), and because the uncertain precision of the counts of the bacteria, it seems necessary to group the counts into larger classes. The first result of interest for us is the value of the k and p parameters of the negative binomial distribution (fitted using the maximum likelihood method), and the estimates of the sample and theoretical mean, variance, skewness and kurtosis. As a conclusion, the presence of the bacteria of interest in the river in which the sample were collected, is follows a negative binomial distribution (k=0.839, p=5.763), with a mean of 4.8 and a variance of 32.7. www.xlstat.com /en/support/tutorials/histo.htm   (761 words)

 Poisson and Negative Binomial Regression This test tests equality of the mean and the variance imposed by the Poisson distribution against the alternative that the variance exceeds the mean. The asymptotic distribution of the LR statistic has probability mass of one half at zero and one half – Chi-sq distribution with 1 df (see A. Cameron, P.K.Trivedi, Regression Analysis of Count Data, Cambridge University Press, 1998). Instead of assuming as before that the distribution of  Y, number of occurrences of an event, is Poisson, we will now assume that Y has a negative binomial distribution. www.uky.edu /ComputingCenter/SSTARS/P_NB_3.htm   (741 words)

 Ed231C: Negative Binomial Models Negative binomial regression is used to estimate count models when the poisson estimation is inappropriate due to overdispersion (which is most of the time). However, in the context of count regression models the negative binomial distribution can be thought of as a poisson distribution with unobserved heterogeneity which, in turn, can be conceptualized as a mixture of two probability distributions, poisson and gamma. It is possible to estimate a generalized version of the negative binomial model. www.gseis.ucla.edu /courses/ed231c/notes1/nbreg1.html   (766 words)

 Geometric Distribution   (Site not responding. Last check: 2007-10-03) The geometric distribution is similar to the binomial distribution in that the probability of occurrence is constant from trial to trial and the trials are independent. The binomial distribution models situations where the number of trials is fixed, and the random variable is the number of successes. The negative binomial distribution models the number of trials required to obtain m successes, and m is not required to be equal to one. www.engineeredsoftware.com /lmar/geometric.htm   (281 words)

 ISE 162 Sec. 1, Class Notes, Class 7   (Site not responding. Last check: 2007-10-03) Don't be confused by N and n; the first is the size of the finite set from which the sample is drawn, the second the size of the sample. The hypergeometric distribution is a model for an engineering process where the population is finite. So, the negative binomial distribution is a model for an engineering process; k and p define the shape of the probability distribution, and X tells us what the values on the abscissa will be. www.engr.sjsu.edu /jgille/notes2005b07.html   (1162 words)

 negative binomial distribution   (Site not responding. Last check: 2007-10-03) The negative binomial distribution is a variation on the binomial distribution. The distribution gives the probability that n experiments are needed to reach N successes. In formula, the negative distribution is related with a factor n/N to the normal binomial distribution. www.2dcurves.com /discrete/discreten.html   (52 words)

 Florida Entomologist, v. 80, n. 1, p. 1 Based on the negative binomial, mean rust mite densities could be estimated from the percentage of samples with at least one mite. The negative binomial probability distribution is characterized by two parameters, the mean (x) and a coefficient k (Johnson and Kotz 1969). The value of the k parameter defines the shape of the negative binomial distribution and serves as a general indicator of aggregation, with smaller values of k indicating increased aggregation (Southwood 1978). www.fcla.edu /FlaEnt/fe80p1.htm   (2777 words)

 [No title] (When r=1, the negative binomial distribution is the same as the geometric distribution--the distribution of the number of draws until a ticket labeled "1" is drawn for the first time.) Suppose X is a random variable with the negative binomial distribution with parameters p and r. The essential requirement for a random variable to have the negative binomial distribution is that it count the number of trials to the rth success in independent trials with the same probability p of success in each trial. The binomial, geometric, hypergeometric, and negative binomial distributions are examples of discrete probability distributions. www.stat.berkeley.edu /users/stark/SticiGui/Text/ch12.htm   (5742 words)

 The Negative Binomial Distribution   (Site not responding. Last check: 2007-10-03) In the binomial random variable, we have a fixed number of trials, say n, and the random variable X is the number of successes that occur. In the negative binomial random variable, we wait until a given number of events (usually more than one) have occurred. These four trials constitute a binomial sequence, that is, we have four trials and want the probability of exactly one success among them. www.rose-hulman.edu /math21st/lectures/NBD/NBD_1.html   (504 words)

 The Negative Binomial Distribution The distribution defined by the density function in Exercise 2 is known as the of the distribution and hence the negative binomial distribution is There is also an easy solution to the problem of points using the negative binomial distribution In a sense, this has to be the case, given the equivalence between the binomial and negative binomial processes. www.ds.unifi.it /VL/VL_EN/bernoulli/bernoulli5.html   (1248 words)

 Lecture 4—Wednesday, January 18, 2006 Comment: Since the Poisson distribution is a probability distribution, it must be the case that when we sum over all possible probabilities we get 1. is said to have a negative binomial distribution with parameter p (and r). The negative binomial is a two-parameter distribution, but like the ordinary binomial one of the parameters, in this case r, is usually treated as known. www.unc.edu /courses/2006spring/ecol/145/001/docs/lectures/lecture4.htm   (1355 words)

 Regression Models for Event Count Data Using SAS, STATA, and LIMDEP Thus, researchers have developed various nonlinear models that are based on the Poisson distribution and negative binomial distribution. As the mean of a Poisson distribution increases, the probability of zeros decreases and the distribution approximates a normal distribution (Figure 1). The negative binomial regression model (NBRM) incorporates observed and unobserved heterogeneity into the conditional mean, mu=exp(xb+e) (Long 1997). www.indiana.edu /~statmath/stat/all/count/count1.html   (848 words)

 Histograms The first result of interest for us is the value of the k and p parameters of the negative binomial distribution (fitted using the maximum likelihood method), and the estimates of the sample and theoretical mean and variance. From the results we obtain (p-value=0.129), with a significance level of 0.05, we conclude that we cannot reject the hypothesis that the counts follow a negatve binomial distribution with parameters k=0.839, p=5.763. This test is known as being better suited than the Chi-square test for the continuous distribution functions, which is not the case with the negative binomial distribution. www.kovcomp.co.uk /support/XL-Tut/demo-histo.html   (967 words)

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