Question

The joint probability distribution for Xi and X2 is T1T2 f (1, 2)L22 for 1 1,2,3 and 2 1,2,3. (a) Find the probability distribution of Y XX2 (b) Find the probability distribution of Z
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Answer #1

a) Y can take the values as 1, 2, 3, 4, 6, 9

Y (X_1, X_2) P(y
1 (1,1) 36
2 (1,2) , (2,1) 36
3 (1,3) , (3,1) 36
4 (2,2) 36
6 (2,3) , (3,2) 12 36 3
9 (3,3) 36

So the PMF is

Y 1 2 3 4 6 9
P(y) 1/36 1/9 1/6 1/9 1/3 1/4

b) Z can take the values as 2,3 32 32

The values with the corresponding probabilities are

Z (X_1,X_2) P(z)
{1over 3} (1,3) 36
{1over 2} (1,2) 2 36
3 (2,3) 36
1 (1,1), (2,2), (3,3) 36
{3over 2} (3,2) 36
2 (2,1) 2 36
3 (3,1) 36

So the probability distribution is

Z 1/3 2 2/3 1 3 /2 2 3
P(z) 1/12 1/18 1/6 7/18 1/6 1/18 1/12
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