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
![](https://test.transtutors.com/qimg/e0802a72-20ff-4ccf-b204-9e53aa889e35.png)
Exercise 2
For positive constants ,a , b, c and positive integer , a pair of random variables has a joint PDF specified by
![](https://test.transtutors.com/qimg/ade0b08a-b469-4413-9429-a145cf27f89c.png)
Exercise 3
For some integer L and constant c, two discrete random variables have a joint PMF given by
![](https://test.transtutors.com/qimg/66e2ff63-a41e-43c8-8861-479c14d6a571.png)
(a) Find the value of the constant c in terms of L .
(b) Find the marginal PMFs, ![](https://test.transtutors.com/qimg/7b294816-fd66-42a5-a838-f308e6d07144.png)
(c) Find ![](https://test.transtutors.com/qimg/1aae2302-4af0-4f6b-8cde-4f258398d717.png)
Exercise 4
Two discrete random variables have a joint PMF as described in the following table.
![641_Table.jpg](https://secure.tutorsglobe.com/CMSImages/641_Table.jpg)
![](https://test.transtutors.com/qimg/3d3c23a4-29b3-438f-9f3b-acfeddec1e94.png)