# Binomial and NegBin distribution

• Apr 3rd 2011, 11:46 AM
davemk
Binomial and NegBin distribution
Hi Folks.

I was hoping someone could help me understand this please.

Basically I need to find a relationship between the PMFs for the above. I can see the obvious similarities between the two PMFs and can explain them but an algebraic relationship eludes me.

I found some info online that says:

fXn(x) = fNx(n) + fMn-x(n)

where Xn is the number of successes in n trials; Nx is the number of trials to get x successes and Mn-x is the number of trials to get (n-x?) failures.

For info, the NegBin I'm interested in is the PMf [(x-1)!/(n-1)!(x-n)!]*p^n*(1-p)^x-n

All help appreciated. Thanks.

Dave.
• Apr 3rd 2011, 04:03 PM
theodds
You might notice that, if you fix the number of successes and the number of trials, the pmfs are proportional. This ties into the so-called Likelihood Principle. Because of this relationship, if you accept the LP it essentially means that whether you observe a negative binomial process or a binomial process you should make the same inference about the success probability.

I'm not sure if this is what is being looked for by your instructor, but the LP is a very important concept (IMHO) and the relationship between the binomial and negative binomial is often how it is introduced.

See example 1 of this for a short discussion of this particular problem (negative binomial and binomial).
• Apr 4th 2011, 02:03 AM
davemk
Thanks for that. I haven't heard of the LP before so I appreciate the links.

That seems to be how I've answered the question, although I've concentrated on the way that the PMFs are "constructed" rather than an actual algebraic relationship.