Re: Mean of Geometric and Negative Binomial distributions

*To*: mathgroup at smc.vnet.net*Subject*: [mg24081] Re: [mg24066] Mean of Geometric and Negative Binomial distributions*From*: Rob Pratt <rpratt at email.unc.edu>*Date*: Fri, 23 Jun 2000 02:26:48 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Gareth, You need to check the definitions of the appropriate random variables. The discrepancy arises from whether you call X or Y a geometric random variable: X = the number of trials necessary to obtain the first success in a sequence of independent Bernoulli trials, each with success probability p Y = the number of failures before the first success in a sequence of independent Bernoulli trials, each with success probability p In other words, X counts the first success, but Y doesn't, so Y = X - 1. Usually, X is called a geometric r.v., and Y is sometimes called a modified geometric r.v. Evidently, Mathematica uses Y while your other sources use X. Note that Mean[Y] = Mean[X - 1] = Mean[X] - 1 = 1 / p - 1 = (1 - p) / p. Also, since X and Y differ by a constant, the variances ought to be the same, as you confirmed. The same discrepancy arises for the negative binomial. (Replace "first" with "rth" in the above definitions.) Rob Pratt Department of Operations Research The University of North Carolina at Chapel Hill rpratt at email.unc.edu http://www.unc.edu/~rpratt/ On Thu, 22 Jun 2000, Gareth J. Russell wrote: > Dear Group, > > Can anyone tell me why Mathematica returns (1-p)/p for > Mean[GeometricDistribution[p]] and n(1-p)/p for > Mean[NegativeBinomialDistribution[p]], when all sources I have to hand (such > as CRC Standard Mathematical Formulae) give these as 1/p and n/p > respectively? > > The variance expressions agree with CRC, it's just the means that are > different. > > Thanks, > > Gareth > > ================================================== > Dr. Gareth J. Russell > > NEW ADDRESS and E-MAIL FROM 1ST SEPTEMBER, 1999 > > Center for Environmental Research and Conservation > MC 5556 > Columbia University > 1200 Amsterdam Avenue > New York, NY 10027, U.S.A. > > Phone: ++1 212 854 5094 > Fax: ++1 212 854 8188 > E-mail: russell at cerc.columbia.edu > WWW: http://web.utk.edu/~grussell (NO CHANGE) > > OLD ADDRESS (AND STILL MY EMPLOYERS) > > Department of Ecology and Evolutionary Biology > University of Tennessee > 569 Dabney Hall > Knoxville, TN 37996-1610, USA > ================================================== > > > >