**6.1**

**6.2** For Exercise 14.3, the values at *k* = 1, 2, 3 are 0.1000, 0.0994, and 0.1333, which are nearly constant. The Poisson distribution is recommended. For Exercise 14.5, the values at *k* = 1, 2, 3, 4 are 0.1405, 0.2149, 0.6923, and 1.3333, which is increasing. The geometric/negative binomial is recommended (although the pattern looks more quadratic than linear).

**6.3** For the Poisson, *λ* > 0 and so it must be *a* = 0 and *b* > 0. For the binomial, *m* must be a positive integer and 0 < *q* < 1. This requires *a* < 0 and *b* > 0 provided −*b*/*a* is an integer ≥ 2. For the negative binomial, both *r* and *β* must be positive so *a* > 0 and *b* can be anything provided *b*/*a* > −1.

The pair *a* = −1 and *b* = 1.5 cannot work because the binomial is the only possibility but −*b*/*a* = 1.5, which is not an integer. For proof, let *p*_{0} be arbitrary. Then *p*_{1} = (−1 + 1.5/1)*p* = 0.5*p* and *p*_{2} = (−1 + 1.5/2)(0.5*p*) = −0.125*p* < 0.

**6.4**

The factors will be positive (and thus *p*_{k} will be positive) provided *p*_{1} > 0, *β* > 0, *r* > −1, and *r* ≠ 0.

To see that the probabilities sum to a finite amount,

The terms of the summand are the pf of the negative binomial ...

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