Question

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a fair six-sided die is rolled. Let W=X+Y.

a. What is the pdf of W?

b What is E(W)?

Answer 1

(a)

Since each of the four outcomes are equally likely so probability of getting any outcome when a four sided die is rolled is 1/4.

Since each of the six outcomes are equally likely so probability of getting any outcome when a six sided die is rolled is 1/6.

Possible number of outcomes when a four sided and a 6 sided die is rolled is 4*6 = 24.

Probability of each outcome is 1 / 24 = 0.0417

Following table shows all the possible outcomes and probability :

X	Y	W=X+Y	P(W=w)
1	1	2	0.04167
1	2	3	0.04167
1	3	4	0.04167
1	4	5	0.04167
1	5	6	0.04167
1	6	7	0.04167
2	1	3	0.04167
2	2	4	0.04167
2	3	5	0.04167
2	4	6	0.04167
2	5	7	0.04167
2	6	8	0.04167
3	1	4	0.04167
3	2	5	0.04167
3	3	6	0.04167
3	4	7	0.04167
3	5	8	0.04167
3	6	9	0.04167
4	1	5	0.04167
4	2	6	0.04167
4	3	7	0.04167
4	4	8	0.04167
4	5	9	0.04167
4	6	10	0.04167

To find the pdf of W we need to combine the same values of W. Following is the pdf of W:

W	P(W=w)
2	0.04167
3	0.08334
4	0.12501
5	0.16668
6	0.16668
7	0.16668
8	0.12501
9	0.08334
10	0.04167

b)

The expected value of W is

W	P(W=w)	W*P(W=w)
2	0.04167	0.08334
3	0.08334	0.25002
4	0.12501	0.50004
5	0.16668	0.8334
6	0.16668	1.00008
7	0.16668	1.16676
8	0.12501	1.00008
9	0.08334	0.75006
10	0.04167	0.4167
Total	1.00008	6.00048

So the expected value is