Let y1 and y2 be independent negative binomial variates with common dispersion parameter ????.

a. Show that y1 + y2 is negative binomial with dispersion parameter ????∕2.

b. Conditional on y1 + y2, show that y1 has a beta-binomial distribution.

c. State the multicategory extension of

(b) that yields a Dirichlet-multinomial distribution. Explain the analogy with the Poisson-multinomial result in Section 7.2.1.