1 4 3 3 pts Suppose X has uniform distribution on 0 1 2 th

1. (4 + 3 + 3 pts.) Suppose X has uniform distribution on {0, 1, 2} think of X as the number shown when you spin a spinner in which 0, 1,2 are all equally likely. After you spin and record the number X, roll X fair dice and record the sum Y (if no dice are rolled, the sum is zero). Find (a) the joint distribution f(x, y) of X and Y (answer is a table with 13 rows and 3 columns) (b) the conditional distribution P(Y = y|X = x) on Y when X = x (answer is either a formula, or three tables showing the distributions when X 0, 1, 2) (c) the marginal distribution on Y, that is, find P(Y y). (table)

Solution

a. The joint distribution of X and Y is

b. The conditional distribution is

For X = 0

For X = 1

For X = 2

c. The marginal distribution of Y is

			X
f(x,y)		0	1	2
	0	1/3	0	0
	1	0	1/18	0
	2	0	1/18	1/108
	3	0	1/18	2/108
	4	0	1/18	3/108
	5	0	1/18	4/108
Y	6	0	1/18	5/108
	7	0	0	6/108
	8	0	0	5/108
	9	0	0	4/108
	10	0	0	3/108
	11	0	0	2/108
	12	0	0	1/108