In: Math
Compute the probability distribution, expectation, and variance of the following random variable:
- Multipliying the result of rolling two dice.
below is the probability distribution of rolling two dice:
x | P(x) |
1 | 1/36 |
2 | 1/18 |
3 | 1/18 |
4 | 1/12 |
5 | 1/18 |
6 | 1/9 |
8 | 1/18 |
9 | 1/36 |
10 | 1/18 |
12 | 1/9 |
15 | 1/18 |
16 | 1/36 |
18 | 1/18 |
20 | 1/18 |
24 | 1/18 |
25 | 1/36 |
30 | 1/18 |
36 | 1/36 |
x | f(x) | xP(x) | x2P(x) |
1 | 1/36 | 0.028 | 0.028 |
2 | 1/18 | 0.111 | 0.222 |
3 | 1/18 | 0.167 | 0.500 |
4 | 1/12 | 0.333 | 1.333 |
5 | 1/18 | 0.278 | 1.389 |
6 | 1/9 | 0.667 | 4.000 |
8 | 1/18 | 0.444 | 3.556 |
9 | 1/36 | 0.250 | 2.250 |
10 | 1/18 | 0.556 | 5.556 |
12 | 1/9 | 1.333 | 16.000 |
15 | 1/18 | 0.833 | 12.500 |
16 | 1/36 | 0.444 | 7.111 |
18 | 1/18 | 1.000 | 18.000 |
20 | 1/18 | 1.111 | 22.222 |
24 | 1/18 | 1.333 | 32.000 |
25 | 1/36 | 0.694 | 17.361 |
30 | 1/18 | 1.667 | 50.000 |
36 | 1/36 | 1.000 | 36.000 |
total | 12.250 | 230.028 | |
E(x) =μ= | ΣxP(x) = | 12.2500 | |
E(x2) = | Σx2P(x) = | 230.0278 | |
Var(x)=σ2 = | E(x2)-(E(x))2= | 79.9653 |
from abvoe expectation =12.25
Variance =79.9653