Question

Consider a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6, where X is the number appearing on the four-sided die and Y is the number appearing on the six-sided die. Define W=X+Y when they are rolled together. Assuming X and Y are independent, (a) find the moment generating function for W, (b) the expectation E(W), (c) and the variance Var(W). Use the moment generating function technique to find the expectation and variance.

Answer 1

Following table shows the different values of X, Y and W and corresponding probabilites:

X	P(X=x)	Y	P(Y=y)	W	P(W=w)=P(X=x)P(Y=y)
1	0.25	1	0.1667	2	0.041675
1	0.25	2	0.1667	3	0.041675
1	0.25	3	0.1667	4	0.041675
1	0.25	4	0.1667	5	0.041675
1	0.25	5	0.1667	6	0.041675
1	0.25	6	0.1667	7	0.041675
2	0.25	1	0.1667	3	0.041675
2	0.25	2	0.1667	4	0.041675
2	0.25	3	0.1667	5	0.041675
2	0.25	4	0.1667	6	0.041675
2	0.25	5	0.1667	7	0.041675
2	0.25	6	0.1667	8	0.041675
3	0.25	1	0.1667	4	0.041675
3	0.25	2	0.1667	5	0.041675
3	0.25	3	0.1667	6	0.041675
3	0.25	4	0.1667	7	0.041675
3	0.25	5	0.1667	8	0.041675
3	0.25	6	0.1667	9	0.041675
4	0.25	1	0.1667	5	0.041675
4	0.25	2	0.1667	6	0.041675
4	0.25	3	0.1667	7	0.041675
4	0.25	4	0.1667	8	0.041675
4	0.25	5	0.1667	9	0.041675
4	0.25	6	0.1667	10	0.041675

Following table shows the MGF of W:

W	P(W=w)
2	0.041675
3	0.08335
4	0.125025
5	0.1667
6	0.1667
7	0.1667
8	0.125025
9	0.08335
10	0.041675
Total	1.0002

(a)

The MGF of W is:

(b)

Differentiating above with respect to t gives:

Putting t=0 gives:

(c)

Differentiating above with respect to t again gives:

Putting t=0 gives:

So variance of W is