Parameter-Mix Distributions

The notation:

[Graphics:Images/mix_gr_1.gif]

denotes a [Graphics:Images/mix_gr_2.gif] distribution in which parameter [Graphics:Images/mix_gr_3.gif] (instead of being fixed) has an [Graphics:Images/mix_gr_4.gif] distribution. We wish to find the unconditional distribution of [Graphics:Images/mix_gr_5.gif]...

Given: The pmf of [Graphics:Images/mix_gr_6.gif] is [Graphics:Images/mix_gr_7.gif]:

[Graphics:Images/mix_gr_8.gif]

Given: The pdf of parameter [Graphics:Images/mix_gr_9.gif] is [Graphics:Images/mix_gr_10.gif]:

[Graphics:Images/mix_gr_11.gif]

Then, the parameter-mix distribution is the expectation [Graphics:Images/mix_gr_12.gif] with respect to the distribution of Θ. The solution with mathStatica is simply:

[Graphics:Images/mix_gr_13.gif]
[Graphics:Images/mix_gr_14.gif]

with domain of support:

[Graphics:Images/mix_gr_15.gif]

This is known as Holla's distribution. Here is a plot of its pmf when [Graphics:Images/mix_gr_16.gif] and [Graphics:Images/mix_gr_17.gif]:

[Graphics:Images/mix_gr_18.gif]

[Graphics:Images/mix_gr_19.gif]

Fig. 1: The pmf of Holla's distribution when [Graphics:Images/mix_gr_20.gif] and [Graphics:Images/mix_gr_21.gif]