Practice Problem Set 12 – (a,b,1) class
The practice problems in this post are to reinforce the concepts of (a,b,0) class and (a,b,1) class discussed this blog post and this blog post in a companion blog. These two posts have a great deal of technical details, especially the one on (a,b,1) class. The exposition in this blog post should be more accessible.
Notation: for whenever is the counting distribution that is one of the (a,b,0) distributions – Poisson, binomial and negative binomial distribution. The notation is the probability that a zerotruncated distribution taking on the value . Likewise is the probability that a zeromodified distribution taking on the value .
Practice Problem 12A 

Practice Problem 12B 
This problem is a continuation of Problem 12A. The following is the probability generating function (pgf) of the Poisson distribution in Problem 12A.

Practice Problem 12C 
Consider a negative binomial distribution with and .

Practice Problem 12D 
The following is the probability generating function (pgf) of the negative binomial distribution in Problem 12C.

Practice Problem 12E 
This is a continuation of Problem 12C and Problem 12D.

Practice Problem 12F 
This problem is similar to Problem 12E.

Practice Problem 12G 
Suppose that the following three probabilities are from a zerotruncated (a,b,0) distribution.

Practice Problem 12H 
Consider a zeromodified distribution. The following three probabilities are from this zeromodified distribution.

Practice Problem 12I 
For a distribution from the (a,b,0) class, you are given that
Determine . 
Practice Problem 12J 
Generate an extended truncated negative binomial (ETNB) distribution with and . Note that this is to start with a negative binomial distribution with parameters and and then derive its zerotruncated distribution. The parameters and will not give a distribution but over look this point and go through the process of creating a zerotruncated distribution. In particular, determine the following.

Problem  Answer 

12A 

12B 

12C 

12D 

21E 

12F 

12G 

12H 

12I 

12J 

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