Jointly distributed random variables

Two discrete random variables Let X and Y be two discrete random variables defined on the same sample space. The joint probability mass function is defined by We must have and . Also .

Example 1 Let X be the deductible on an auto policy and Y the deductible on a homeowner’s policy for a particular company. The possible deductibles are $100 and $250 on auto policies, and $0, $100 and $200 on homeowners policies. The joint pmf is: 100 200 .20 .10 250 .05 .15 .30

Example 1 (continued) Then ,

Definition of marginal probability mass function The marginal probability distributions are defined as for each possible x for each possible y

Marginals for example 1 100 200 .20 .10 .5 250 .05 .15 .30 .25 .50

Two continuous random variables Let X and Y be two continuous rv’s. The joint probability density function f(x,y) is a function satisfying and . Then .

Marginal densities The marginal density functions of X and Y are

Example 2 A nut company markets cans of deluxe mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly 1 lb., but the composition can vary. Let X=weight of almonds and Y=weight of cashews. Let the joint density be

Example 2 (continued) This is a valid density (verify), and

Independent random variables Two rv’s are independent if for every pair of x and y values (discrete) or (continuous)

Example 1 revisited For the insurance example, so X and Y are not independent.

Example 2 revisited For the mixed nuts, so X and Y are not independent. To be independent, the density must have the form and the region of positive density must be a rectangle whose sides are parallel to the coordinate axis.

Extension to more than two variables If are discrete rv’s, the joint pmf is the function For continuous rv’s, the joint density is the function such that for n intervals ,

Extension of definition of independence The random variables are independent if for every subset of the variables (each pair, each triple, and so on), the joint pmf or pdf of the subset is equal to the product of the marginal pmf’s or pdf’s for all possible values of the variables.

Conditional distributions Let X and Y be two continuous rv’s with joint pdf f(x,y). Then for any x such that , the conditional probability density function of Y given that X=x is , . For X and Y discrete, replacing the pdf’s by pmf’s gives the conditional probability mass function of Y when X=x.

Example 2 again For the mixed nuts example,