a) All the possible outcomes of n = 2 are:
| Customer 1 |
Customer 2 |
Mean |
R |
Probability |
| 1 |
1 |
1 |
0 |
0.01 |
| 1 |
2 |
1.5 |
1 |
0.04 |
| 1 |
3 |
2 |
2 |
0.02 |
| 1 |
4 |
2.5 |
3 |
0.03 |
| 2 |
1 |
1.5 |
1 |
0.04 |
| 2 |
2 |
2 |
0 |
0.16 |
| 2 |
3 |
2.5 |
1 |
0.08 |
| 2 |
4 |
3 |
2 |
0.12 |
| 3 |
1 |
2 |
2 |
0.02 |
| 3 |
2 |
2.5 |
1 |
0.08 |
| 3 |
3 |
3 |
0 |
0.04 |
| 3 |
4 |
3.5 |
1 |
0.06 |
| 4 |
1 |
2.5 |
3 |
0.03 |
| 4 |
2 |
3 |
2 |
0.12 |
| 4 |
3 |
3.5 |
1 |
0.06 |
| 4 |
4 |
4 |
0 |
0.09 |
The probability distribution is:
| X |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
| P(X) |
0.01 |
0.08 |
0.2 |
0.22 |
0.28 |
0.12 |
0.09 |
b) P(X <= 2.5) = 0.01 + 0.08 + 0.20 + 0.22 =
0.51
c) The probability distribution from part a) is:
| R |
0 |
1 |
2 |
3 |
| P (R) |
0.3 |
0.36 |
0.28 |
0.06 |
d) Let's just list the outcomes with X <= 1.5:
| C1 |
C2 |
C3 |
C4 |
Mean |
Probability |
| 1 |
1 |
1 |
1 |
1 |
0.0001 |
| 2 |
1 |
1 |
1 |
1.25 |
0.0004 |
| 1 |
2 |
1 |
1 |
1.25 |
0.0004 |
| 1 |
1 |
2 |
1 |
1.25 |
0.0004 |
| 1 |
1 |
1 |
2 |
1.25 |
0.0004 |
| 3 |
1 |
1 |
1 |
1.5 |
0.0002 |
| 1 |
3 |
1 |
1 |
1.5 |
0.0002 |
| 1 |
1 |
3 |
1 |
1.5 |
0.0002 |
| 1 |
1 |
1 |
3 |
1.5 |
0.0002 |
| 2 |
2 |
1 |
1 |
1.5 |
0.0016 |
| 2 |
1 |
2 |
1 |
1.5 |
0.0016 |
| 2 |
1 |
1 |
2 |
1.5 |
0.0016 |
| 1 |
2 |
2 |
1 |
1.5 |
0.0016 |
| 1 |
2 |
1 |
2 |
1.5 |
0.0016 |
| 1 |
1 |
2 |
2 |
1.5 |
0.0016 |
The probability is: 0.0121
