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Q on Type 2 Negative Binomial Parameters

D

DMF

Member
I was working through a question in one of the assignments and in an alternative solutions, it is indicated that 2/[3^(n+1)] can be considered a Type 2 Neg. Bin. distribution with p=2/3 , q =1/3 and k=1. I am not sure how these parameters are defined. Any insight / hints would be appreciated.
Thanks,
 
I think you're referring to the mixture distribution question, in which case since there are only 4 cases that "work" I would've remembered that Poi/Gamma combination gives a NBin.

As for spotting it, we know that for a type 2 NBin we have P(N=n) = const * p^k * q^n

We can see we have 3^(n+1) on the denominator, so the 3^n we be the denominator or q^n - we don't know yet whether it will be 1 or 2 on the numerator, but we do know that p will have the other 3 on the denominator.

This tells us that k=1.

So the constant will now be \(\frac{\Gamma(n+1)}{\Gamma(n+1)\Gamma(1)} = 1\)

S0 all we have left is the 2 to go on one of the numerators. It must be the p as there is only one of them.
 
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