Examples

Expectation of X squared plus Y from joint PMF chart

Suppose the joint PMF of and is given by this chart:

	0.2	0.2
	0.35	0.1
	0.05	0.1

Define . Find the expectation .

Solution

First compute the values of for each pair in the chart:

	0	3
	1	4
	2	5

Now take the sum, weighted by probabilities:

Expectation of Y two ways and Expectation of XY from joint density

Suppose and are random variables with the following joint density:

(a) Compute using two methods.

(b) Compute .

Solution

(a)

(1) Method One: via marginal PDF :

Then expectation:

(2) Method Two: directly, via two-variable formula:

(b) Directly, via two-variable formula: