denotes a distribution in which parameter (instead of being fixed) has an distribution. We wish to find the unconditional distribution of ...
Given: The pmf of is :
Given: The pdf of parameter is :
Then, the parameter-mix distribution is the expectation with respect to the distribution of Θ. The solution with mathStatica is simply:
with domain of support:
This is known as Holla's distribution. Here is a plot of its pmf when and :
Fig. 1: The pmf of Holla's distribution when and