W10 - Examples

Functions on two random variables

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:

two ways, and , from joint density

Suppose and are random variables with the following joint density:

(a) Compute using two methods.

(b) Compute .

Solution (a) Method One: via marginal PDF :

Then expectation:

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

(b) Directly, via two-variable formula:

Covariance from 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
Find .

Solution We need and and .

Therefore:

Variance of sum of indicators

An urn contains 3 red balls and 2 yellow balls.

Suppose 2 balls are drawn without replacement, and counts the number of red balls drawn.

Find .

Solution Let indicate (one or zero) whether the first ball is red, and indicate whether the second ball is red, so .

Then indicates whether both drawn balls are red; so it is Bernoulli with success probability . Therefore .

We also have .

The variance sum rule gives: