Question

In: Statistics and Probability

1. Suppose that you roll two dice simultaneously. Let X be the random variable that gives...

1. Suppose that you roll two dice simultaneously. Let X be the random variable that gives the product of the two die outcomes. Let f(x) = P(X = x) be the probability distribution for X.

(a) Is X discrete or continuous?

(b) What is f(12)?

(c) If F(x) is the cumulative distribution function, what is F(4)?

Expert Solution

The exhaustive cases of two times throwing of a die is (Sample Space)

S = { (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)}

n = 36

Let X be the random variable that gives the product of the two die outcomes

a) X is discrete random variable

since it represents countable values

The PMF of X is

X : 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 30 36

P(X=x) : 1/36 2/36 2/36 3/36 2/36 4/36 2/36 1/36 2/36 4/36 2/36 1/36 2/36 2/36 2/36 1/36 2/36 1/36

b) f(12) = 3/36

Sets of multiple of 4 is { (1,4), (4,1), (2,2) } The number of pairs is 3

c) The cumulative distribution function of X is

X : 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 30 36

P(X=x) : 1/36 2/36 2/36 3/36 2/36 4/36 2/36 1/36 2/36 4/36 2/36 1/36 2/36 2/36 2/36 1/36 2/36 1/36

F(x) : 1/36 3/36 5/36 8/396 10/36 14/36 16/36 17/36 19/36 23/36 25/36 26/36 28/36 30/36 32/36 33/36 35/36 36/36

F(4) = 8/36

orchestra answered 10 months ago

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