I've got this problem on multivariable distributions I'm stuck on...
Let denote a random sample of size n = 3 from a distribution with the geometric pdf
and given the distribution
Question find the pdf of then find .
Now I considered using the moment generating function for 3 reasons: (a) are a random sample size are independent (b) they have the same pdf thus mgf, and (c) well, mgf can be used for this.
So i started out like this:
Then I got stuck there... I presume I don't know how to "transform" variables in descrete distributions unlike continuous ones.
So any help is much appreciated!
Try defining the geometrical distribution as
That should get rid of the exponential in the denominator of the MGF. Then I think you will have a match with your definition of the negative binomial.
Another possible source of confusion is whether p is the probability of success or the probability of failure. Again, you need a consistent set of definitions in order for everything to work out.