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**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)

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#### Solution

**a.** ∵ X has probability distribution,

∴ ∑p_{i} = 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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