In the setting of examine whether the normalizing constant of the geometric average of the Poisson and negative binomial distributions,   can be computed.

When comparing two models M₁ and M₂, with densities    and   both from an exponential family,

a. Show that the geometric average

still belongs to an exponential family.

b. Show that, if (i = 1, 2) 

  is a sufficient statistic for the geometric average.

c. Deduce that, if  is of full rank, the dimension of this family (see Definition 3.3.2) is the sum of the dimensions of f₁ and f₂.

d. In the special case when M₁ is the exponential  is the half-normal  model, show that the geometric average model is the half-normal distribution.

and give its normalizing constant.