1. (4 + 3 + 3 pts.) Suppose X has uniform distribution on {0, 1, 2} think of X a

Question

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)

Explanation / Answer

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

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