Question

In: Statistics and Probability

Two dice are rolled. Let the random variable X denote the number that falls uppermost on...

Two dice are rolled. Let the random variable X denote the number that falls uppermost on the first die and let Y denote the number that falls uppermost on the second die.

(a) Find the probability distributions of X and Y.

x	1	2	3	4	5	6
P(X = x)

y	1	2	3	4	5	6
P(Y = y)

(b) Find the probability distribution of X + Y.

x + y	2	3	4	5	6	7
P(X + Y = x + y)

x + y	8	9	10	11	12
P(X + Y = x + y)

Expert Solution

a)

x P(X = x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

y P(Y = y)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

b)

x+y P(X + Y = x+y)

2 P(1,1) = 1/6 * 1/6 = 1/36

3 P(1,2) + P(2,1) = 2 * 1/6 * 1/6 = 1/18

4 P(1,3) + P(2,2) + P(3,1) = 3 * 1/6 * 1/6 = 1/12

5 P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 * 1/6 * 1/6 = 1/9

6 P(1,5) + P(2,4) + P(3,3) + P(4,2) + P(5,1) = 5 * 1/6 * 1/6 = 5/36

7 P(1,6) + P(2,5) + P(3,4) + P(4,3) + P(5,2) + P(6,1) = 6 * 1/6 * 1/6 = 1/6

8 P(2,6) + P(3,5) + P(4,4) + P(5,3) + P(6,2) = 5 * 1/6 * 1/6 = 5/36

9 P(3,6) + P(4,5) + P(5,4) + P(6,3) = 4 * 1/6 * 1/6 = 1/9

10 P(4,6) + P(5,5) + P(6,4) = 3 * 1/6 * 1/6 = 1/12

11 P(5,6) + P(6,5) = 2 * 1/6 * 1/6 = 1/18

12 P(6,6) = 1/6 * 1/6 = 1/36

orchestra answered 1 year ago

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