Two 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. Find the PMF of Z. Find the joint PMF for X and Z. Find E[XZ]. Find the covariance. Find the correlation coefficient rho X,Z = Cov(X, Z)/ Var(X) Var(Z).

Solution

(a)

The value of Z can be calculated using all the possible outcomes as shown below-

From the table, we can get the frequency distribution of Z as-

As there are 36 possible cases, the probability distribution function of Z can be written as -

P(Z) = Frequency (Z) / 36.

So, we get the following PDF of Z-

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(b)

As for each \'X the distribution of \'Z\' is unique, and the value of Z depends upon the value of X

X	Y	Z
1	1	0
1	2	-1
1	3	-2
1	4	-3
1	5	-4
1	6	-5
2	1	1
2	2	0
2	3	-1
2	4	-2
2	5	-3
2	6	-4
3	1	2
3	2	1
3	3	0
3	4	-1
3	5	-2
3	6	-3
4	1	3
4	2	2
4	3	1
4	4	0
4	5	-1
4	6	-2
5	1	4
5	2	3
5	3	2
5	4	1
5	5	0
5	6	-1
6	1	5
6	2	4
6	3	3
6	4	2
6	5	1
6	6	0