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Question
The probability distribution of X is as follows:
| x | 0 | 1 | 2 | 3 | 4 |
| P[X = x] | 0.1 | k | 2k | 2k | k |
Find
- k
- P[X < 2]
- P[X ≥ 3]
- P[1 ≤ X < 4]
- P(2)
Solution
a. The table gives a probability distribution and therefore P[X = 0] + P[X = 1] + P[X = 2] + P[X = 3] + P[X = 4] = 1
i.e., 0.1 + k + 2k + 2k + k = 1
i.e., 6k = 0.9
∴ k = 0.15
k = 0.15
b. P[X < 2] = P[X = 0] + P[X = 1] = 0.1 + k
= 0.1 + 0.15
= 0.25
c. P[X ≥ 3] = P[X = 3] + P[X = 4] = 2k + k
= 3k
= 3(0.15)
= 0.45
d. P[1 ≤ X < 4] = P[X = 1] + P[X = 2] + P[X = 3]
= k + 2k + 2k
= 5k
= 5(0.15)
= 0.75
e. P(2) = P[X ≤ 2] = P[X = 0] + P[X = 1] + P[X = 2]
= 0.1 + k + 2k
= 0.1 + 3k
= 0.1 + 0.45
= 0.55
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