Examples

Minimal MSE estimate given PMF

Suppose has the following PMF:

	1	2	3	4	5
	0.15	0.28	0.26	0.19	0.13

Find the minimal MSE estimate of , given that is even. What is the error of this estimate?

Solution

The minimal MSE given is just where .

First compute the conditional PMF:

Therefore:

The error is:

Suppose that and suppose .

(a) Find (b) Find (c) Find

Solution

(a) Find .

We know .

Given , so is uniform on , we have .

(b) Find .

(1) We know .

To compute this function, we calculate a sequence of densities.

(2) We know and . From these we derive the joint distribution :

Now extract the marginal :

Now deduce the conditional :

(3) Then:

So .

(1) We need all the basic statistics.

because .

using the marginal PDF on . (IBP and L’Hopital are needed.)

also using the marginal .

(2) using , namely:

From this we infer and .

(3) Hence:

Thus: