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

1. Determine the value of k.
2. Find P(X ≤ 4)
3. P(2 < X < 4)
4. P(X ≥ 3)

Answer 1

a. ∵ X has a 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`