Determine whether or not each of the following pairs of random variables are independent:

(a) The random variables described in Exercise 1;

(b) The random variables described in Exercise 2;

(c) The random variables described in Exercise 3;

(d) The random variables described in Exercise 4.

Exercise 1

For positive constants and , a pair of random variables has a joint PDF specified by

Exercise 2

For positive constants ,a , b, c and positive integer , a pair of random variables has a joint PDF specified by

Exercise 3

For some integer L and constant c, two discrete random variables have a joint PMF given by

(a) Find the value of the constant c in terms of L .

(b) Find the marginal PMFs, 

(c) Find 

Exercise 4

Two discrete random variables have a joint PMF as described in the following table.