Mixture Distributions

The notation:

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 :

[Graphics:Images/mix_gr_19.gif]

Fig. 1: The pmf of Holla's distribution when and