Question

1 Let f(x, y) = 4xy, 0 < x < 1, 0 < y < 1, zero elsewhere, be the joint probability density function(pdf) of X and Y . Find P(0 < X < 1/2 , 1/4 < Y < 1) , P(X = Y ), and P(X < Y ). Notice that P(X = Y ) would be the volume under the surface f(x, y) = 4xy and above the line segment 0 < x = y < 1 in the xy-plane.

2. Let X and Y be two discrete random variables and the joint probability mass function (joint distribution) of X and Y is given by the following table:

1 2 3 4 (X)

1 0.10 0.05 0.02 0.02

2 0.05 0.20 0.05 0.02

3 0.02 0.05 0.20 0.04

4 0.02 0.02 0.04 0.10

(Y)

(a) Find the marginal distribution of X, i.e. construct a table such as

Values of X 1 2 3 4

Probabilities ? ? ? ?

Repeat the same for Y .

(b) Are X and Y independent? Why or Why not?

3. Let X and Y have the following joint density f(x, y): f(x, y) = e^−x if x > 0 and 0 < y < 1, 0 otherwise 1

(a) What are the marginal distributions of X and Y ? In other words, find the density of X, given by fX(x), and the density of Y , given by fY (y).

(b) Are X and Y independent?

(c) What is P(X^2 > 25, Y < 0.5)?

Answer 1