In: Statistics and Probability
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)?
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