Roll two fair six-sided dice, and let X, Y denote the first and the second numbers....

Question

Question

Roll two fair six-sided dice, and let X, Y denote the first and the second numbers.

If Z=max {X, Y}, find

- E(Z)

- V(Z)

If Z=|X-Y|, find

- E(Z)

- V(Z)

Roll two fair six-sided dice, and let X, Y denote the first and the second numbers. If Z max X, Y), find . E(Z) . V(Z) . E(Z) V(Z)

math Statistics-And-Probability

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Answer 1

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Answer #1

The possible values of (X,Y) are

{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Thus, Distribution of Z = max{X,Y} is

Z	1	2	3	4	5	6
P(Z)	1/36	3/36	5/36	7/36	9/36	11/36

Thus, E(Z) = sum z.P(z)

= 161/36

= 4.472

V(Z) = $E(Z^{2})-E(Z)^{2}$

= 2. P(z) 4.47 2

= 791/36 - 20

= 1.972

Now, Distribution of Z = I X - Y I is

Z	0	1	2	3	4	5
P(Z)	6/36	10/36	8/36	6/36	4/36	2/36

hus, E(Z) = sum z.P(z)

= 70/36

= 1.9444

V(Z) = $E(Z^{2})-E(Z)^{2}$

= 2P(z) - 1.944

= 2.0525

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Answer 2

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