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Question
The probability distribution of a discrete r.v. X is as follows:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 4)
- P(X ≥ 3)
Solution
a. ∵ X has probability distribution,
∴ ∑pi = 1
⇒ k + 2k + 3k + 4k + 5k + 6k = 1
⇒ 21k = 1
⇒ k = `1/21`
b. P(X ≤ 4) = P(X = 1) + P(X = 2) + P
(X = 3) + P(X = 4)
= k + 2k + 3k + 4k = 10k
= `10 xx 1/21`
= `10/21`
c. P(2 < X < 4) = P(X = 3)
= 3K
= `3/21`
= `1/7`
d. P(X ≥ 3) = 1 – P(X < 3)
= 1 – [P(X = 1) + P(X = 2)]
= 1 – [k + 2k]
= 1 – 3k
= `1 - 3/21`
= `18/21`
= `6/7`
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