Homework SourseTwo fair dice are rolled and the outcomes are mapped to X an
Two fair dice are rolled, and the outcomes are mapped to X and Y . A third random
variable Z = X Y is computed.
(a) Find the PMF of Z.
(b) Find the joint PMF for X and Z.
(c) Find E[XZ].
(d) Find the covariance.
(e) Find the correlation coefficient pX,Z = Cov(X,Z)/Var(X)Var(Z).
Solution
Two fair dice are rolled ,the outcomes are mapped in X and Y
The outcomes for X ={1,2,3,4,5,6}
The outcomes for Y ={1,2,3,4,5,6}
value of Z=X-Y is between -5 to 5 with probability 1
| Z | Favourable sample point | No. of favourable cases | PMF OF Z =P(Z=X-Y) |
|---|---|---|---|
| -5 | (1,6) | 1 | 1/36 |
| -4 | (1,5) (2,6) | 2 | 2/36 |
| -3 | (1,4) (2,5) (3,6) | 3 | 3/36 |
| -2 | (1,3) (2,4) (3,5) (4,6) | 4 | 4/36 |
| -1 | (1,2) (2,3) (3,4) (4,5) (5,6) | 5 | 5/36 |
| 0 | (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) | 6 | 6/36 |
| 1 | (2,1) (3,2) (4,3) (5,4) (6,5) | 5 | 5/36 |
| 2 | (3,1)(4,2) (5,3)(6,4) | 4 | 4/36 |
| 3 | (4,1)(5,2)(6,3) | 3 | 3/36 |
| 4 | (5,1)(6,2) | 2 | 2/36 |
| 5 | (6,1) | 1 | 1/36 |
