Re: Is n a parameter of a binomial distribution?

In article <1192797455.065283.3030@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<vontressms@xxxxxx> wrote:
On Oct 18, 1:23 pm, Vinayak Rao <vinayak....@xxxxxxxxx> wrote:
Hi,
Is the number of experiments 'n' a parameter of the binomial
distribution

p(k|n,p) = nCk p^k (1-p)^(n-k) ?

The answer would seem to be 'yes' almost by definition, but if I
express the binomial distribution in the "exponential family form", it
has only 1 natural parameter log(p/(1-p)). If n were a parameter,
would this mean that the binomial belongs to the exponential family
only w.r.to parameter p (and not n)?

Vinayak

Vinayak,
n is not usually considered a parameter in the binomial
distribution. It is a fixed number by the design of the experiment. We
usually estimate p in a given number of n trials where there are k
successes, and k is a random variable ranging between zero and n.

Your expression of the density as a member of the exponential family
is another reason why n is not usually considered a "parameter".

Mark


Anything which can be computed from a probability
distribution is a parameter, so n is a parameter.
There is considerable information on the estimation
of n if p is known, and also if p is unknown.

A set of such parameters can be called the parameters
of a family of distributions if their values determine
the distribution. Usually, n and p are the parameters.

The binomial distribution belongs to the exponential
family with respect to p, but not with respect to n.

One gets a rather interesting estimation problem if
the parameters used are h = 1/n and np. In this case,
one gets the binomial distribution if 1/h is a positive
integer, Poisson if h = 0, and negative binomial if h < 0.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

Relevant Pages

  • Re: Faster than light signalling via an EPR type mechanism
    ... The final distribution is still likely to be a binomial ... from that predicted by the binomial distribution (of sq. ... There is a 50 / 50 chance of either photon being up or down, ... become re-aligned with the angle of the filter. ...
    (sci.physics.research)
  • Re: A challenge for Pitman-math
    ... So the chance of getting one head in two tosses would be ... Now back to your example with one tail in 10 tosses. ... Anybody who has had a decent introductory course in probability will tell you that the number of heads in a given number of tosses is governed by a binomial distribution. ...
    (talk.origins)
  • Discrete Probability Dangers 2
    ... The binomial distribution is quite common among discrete distributions. ... if a factory produces 7% defective items independently ...
    (sci.physics)
  • Re: X-Binomial distribution
    ... > variations of the Binomial distribution. ... which illustrates for a Poisson-InverseGaussian parameter mix ... Dr Colin Rose ...
    (sci.stat.math)
  • Re: Help with BUGS model
    ... With the model where the Lambdas are drawn from only one gamma prior I ... sure why your MLE estimation procedure had problems. ... the expected value of the cell under the assumption of independence of ... The posterior distribution of lambda given N has a closed form. ...
    (sci.stat.math)