87. The joint cumulative distribution function of two random variables X and Y, denoted by F(x, y), is de ned by

a. Suppose that X and Y are both continuous variables.

Once the joint cdf is available, explain how it can be used to determine the probability P[(X, Y)  A], where A is the rectangular region

{(x, y): a x

b, c y

d}.

b. Suppose the only possible values of X and Y are 0, 1, 2, . . . and consider the values a5, b10, c2, and d6 for the rectangle speci ed in (a).

Describe how you would use the joint cdf to calculate the probability that the pair (X, Y) falls in the rectangle. More generally, how can the rectangular probability be calculated from the joint cdf if

a, b,

c, and d are all integers?

c. Determine the joint cdf for the scenario of Example 5.1. Hint: First determine F(x, y) for x 

100, 250 and y  0, 100, and 200. Then describe the joint cdf for various other (x, y) pairs.

d. Determine the joint cdf for the scenario of Example 5.3 and use it to calculate the probability that X and Y are both between .25 and .75. Hint:

For 0 x

1 and 0 y

1, F1x, y2  x 0 y 0

f1u, v2 dvdu

e. Determine the joint cdf for the scenario of Example 5.5. Hint: Proceed as in (d), but be careful about the order of integration and consider separately (x, y) points that lie inside the triangular region of positive density and then points that lie outside this region.