# Thread: Binomial and NegBin distribution

1. ## 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.

2. 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).

3. 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.