In: Statistics and Probability
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) |
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