Show that each of the following is an exponential family whose
pdf may be put in the form
p(x|th) = h(x) eT(x) · nu(th) - B(th)
by explicitly identifying:
- Space X where random variable takes values, and dimension q
of Euclidean space Rq containing X
- h(x) (real-valued function on X)
- k, dimension of the sufficient statistic & natural parameter:
- T(x) (Rk-valued Sufficient Statistic)
- nu(th) (Rk-valued Natural Parameter)
- B(th) (real-valued function of Theta)
normal (both mean mu and variance sigma² unknown),
poisson (both mean mu and variance sigma² unknown),
beta (both shape parameters alpha and beta unknown),
gamma (both shape alpha and rate lambda unknown),
binomial (only success prob p unknown, we know N).
Note that some of these are also Exponential Families with some of the
parameters fixed, usually with smaller k-- e.g. the Normal with
known variance has k=1. You may find formulas for the pdf's for
these and many other distributions here.