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
>
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>
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>
>
>

```

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