Negative Binomial Distribution

[kartik_newpro]

Can someone outline the differences between Type I and Type II Negative Binomial Distribution and how to identify which is it?

I would be grateful if it is done both in the theoretical and practical sense (with example).

Thanks.

 

[John Lee]

Can someone outline the differences between Type I and Type II Negative Binomial Distribution and how to identify which is it?

I would be grateful if it is done both in the theoretical and practical sense (with example).

Thanks.

Type 1 counts the number of trials up to and including the kth success.

Type 2 counts the failures before the kth success.

So suppose we are counting the number of goals we score (success) for the penalty kicks we make (trials).

Suppose we score our 3rd goal on our 10th penalty kick.

Under Type 1 X=10 as it is the 10th kick when we got our 3rd goal.

Under Type 2 X=7 as we had 7 failures before we got our 3rd goal.

In CT3 our default choice will be a type 1 (for CT6 it will be type 2).

 

[stylz]

Hi,
Following on from this, in the exam type question on page 13, chapter 7, it specifies a negative binomial distribution for N. In the answer it says that it is a type 2 negative binomial distribution.

I'll have a go at writing the pdf here, but it's hard to get the format right in these text boxes.

The first part is an nCr format, where n = n+2, and r = n. This is then multiplied by 0.9^3x0.1^n.

Might be easier if you just look at the notes yourself to see the pdf clearly (or someone can tell me how to write the formula properly in this text box).

My question is, how is this easily identified as a type 2? My initial thought was type 1. Cheers

 

[didster]

couple of pointers

if n=0,1,2,.. then it's number of failures so type 2; n=k,k+1,... =>type 1.

in your formula
the probabilities have n failures : (P(failure))^n
n+2 things being arranged: again hints at n failures, final success is fixed at end and n+2 other results to arrange.

 

[stylz]

Ok that help didster. Thanks. But what you are saying is in contrast to the book which says type 2....??

 

[didster]

No, in your example, n represents the number of failures, so type 2
Look at it again.