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MathGroup Archive 2000

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Re: Mean of Geometric and Negative Binomial distributions

  • To: mathgroup at
  • Subject: [mg24087] Re: [mg24066] Mean of Geometric and Negative Binomial distributions
  • From: "Richard Finley" <rfinley at>
  • Date: Fri, 23 Jun 2000 02:26:55 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

With the geometric distribution it is because there is a BUG...with the negative
binomial, it is a misunderstanding.  Your reference formulae are correct. Clearly
the problem with Mathematica's implementation of the Geometric Distribution
is that they have neglected the fact that you cannot have a success if there are
no trials (ie if n=0) but in their case PDF[GeometricDistribution[p],0] = p
instead of DF[GeometricDistribution[p],1] = p.  So since it is off by one you can
get the correct mean by calculating
Sum[ n PDF[GeometricDistribution[p], n-1], {n,1,Infinity}]  which gives
1/p as it should.  

For the negative binomial distribution, there is no bug ...but only appears so in the
sense that they calculate a nonstandard result....that is the mean of the total
number of "failures" to get r successes rather than the number of "trials" to get
r successes. You can see this by calculating the results yourself directly rather
than using Mean....

"mean number of trials to get r successes"....
Sum[ (n+r) PDF[NegativeBinomialDistribution[r,p], n] , {n,0,Infinity}]  = r/p

"mean number of failures before getting r successes"....
Sum[ n PDF[NegativeBinomialDistribution[r,p], n], {n, 0, Infinity}] = r (1-p)/p

(Note that I use r for the number of successes and n is the number of failures as opposed to the n in your post which is the number of successes).  

regards,   RF

>>> "Gareth J. Russell" <russell at> 06/21/00 11:01PM >>>
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

The variance expressions agree with CRC, it's just the means that are


Dr. Gareth J. Russell


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 


Department of Ecology and Evolutionary Biology
  University of Tennessee
    569 Dabney Hall
      Knoxville, TN 37996-1610, USA

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