Solved: Suppose that X and Y have a continuous joint

Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is f (x, y) =  k for a x b and c y d, 0 otherwise, where a 0. Find the marginal distributions of X and Y .

Suppose that X and Y have a discrete joint distribution for which the joint p.f. is defined as follows: f (x, y) = 1 30 (x + y) for x = 0, 1, 2 and y = 0, 1, 2, 3, 0 otherwise. a. Determine the marginal p.f.s of X and Y . b. Are X and Y independent?

Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is defined as follows: f (x, y) =  3 2 y2 for 0 x 2 and 0 y 1, 0 otherwise.

Suppose that the joint p.d.f. of X and Y is as follows: f (x, y) =  15 4 x2 for 0 y 1 x2, 0 otherwise. a. Determine the marginal p.d.f.s of X and Y . b. Are X and Y independent?

A certain drugstore has three public telephone booths. For i = 0, 1, 2, 3, let pi denote the probability that exaactly i telephone booths will be occupied on any Monday evening at 8:00 p.m.; and suppose that p0 = 0.1, p1 = 0.2, p2 = 0.4, and p3 = 0.3. Let X and Y denote the number of booths that will be occupied at 8:00 p.m. on two independent Monday evenings. Determine: (a)

Suppose that in a certain drug the concentration of a particular chemical is a random variable with a continuous distribution for which the p.d.f. g is as follows: g(x) =  3 8 x2 for 0 x 2, 0 otherwise. Suppose that the concentrations X and Y of the chemical in two separate batches of the drug are independent random variables for each of which the p.d.f. is g. Determine (a)the joint p.d.f. ofX and Y ;(b)Pr(X = Y );(c)Pr(X > Y ); (d) Pr(X + Y 1

Suppose that the joint p.d.f. of X and Y is as follows: f (x, y) =  2xey for 0 x 1 and 0

Suppose that the joint p.d.f. of X and Y is as follows: f (x, y) =  24xy for x 0, y 0, and x + y 1, 0 otherwise. Are X and Y independent?

Suppose that a point (X, Y ) is chosen at random from the rectangle S defined as follows: S = {(x, y) : 0 x 2 and 1 y 4}. a. Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y . b. Are X and Y independent?

Suppose that a point (X, Y ) is chosen at random from the circle S defined as follows: S = {(x, y):x2 + y2 1}. a. Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y . b. Are X and Y independent?

Suppose that two persons make an appointment to meet between 5 p.m. and 6 p.m. at a certain location, and they agree that neither person will wait more than 10 minutes for the other person. If they arrive independently at random times between 5 p.m. and 6 p.m., what is the probability that they will meet?

Prove Theorem 3.5.6.

In Example 3.5.10, verify that X and Y have the same marginal p.d.f.s and that f1(x) =  2kx2(1 x2)3/2/3 if 1 x 1, 0 otherwise

For the joint p.d.f. in Example 3.4.7, determine whether or not X and Y are independent.

A painting process consists of two stages. In the first stage, the paint is applied, and in the second stage, a protective coat is added. Let X be the time spent on the first stage, and let Y be the time spent on the second stage. The first stage involves an inspection. If the paint fails the inspection, one must wait three minutes and apply the paint again. After a second application, there is no further inspection. The joint p.d.f. of X and Y is f (x, y) = 1 3 if 1 0. Note that it is not exactly a rectangle. b. Find the marginal p.d.f.s of X and Y . c. Show that X and Y are independent. This problem does not contradict Theorem 3.5.6. In that theorem the conditions, including that the set where f (x, y) > 0 be rectangular, are sufficient but not necessary