Examples

Joint and marginal PMFs - Smaller and bigger roll

Roll two dice, and let indicate the smaller of the numbers rolled, and let indicate the bigger number.

Make a chart showing the PMF. Compute the marginal probabilities, and write them in the margins of the chart.

Solution

Event probabilities by reading PMF table

Here is a joint PMF table:

Using the table, compute the following event probabilities: (a) (b) (c) (d)

Joint and marginal PMFs - Coin flipping

Flip a fair coin four times. Let measure the number of heads in the first two flips, and let measure the total number of heads.

Make a chart showing the PMF. Compute the marginal probabilities, and write them in the margins of the chart.

Marginal and event probability from joint density

Suppose the joint density of and is given by:

Find and .

Solution

Compute the marginal PDF:

Find probability of the event :

Marginals from joint density

The joint PDF for and is given by:

Find and .

Event probability from joint density

The joint PDF for and is given by:

Compute .

Properties of joint CDFs

(a) Show with a drawing that if both and , we know:

(b) Explain why:

(c) Explain why: