(Solution) - For each family in Exercise 3 28 describe the natural parameter -(2025 Original AI-Free Solution)

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For each family in Exercise 3.28, describe the natural parameter space.
In Exercise 3.28
Show that each of the following families is an exponential family.
(a) Normal family with either parameter ? or ? known
(b) Gamma family with either parameter ? or ? known or both unknown
(c) Beta family with either parameter ? or ? known or both unknown
(d) Poisson family
(e) Negative binomial family with r known, 0 < p < 1