Question

The random variable X has probability distribution P(X) of the following form, where k is some number:

`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`

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

The random variable X has a probability distribution P(X) of the following form, where 'k' is some real number:

P(X) = `{(k","   if x = 0),(2k"," if x =1),(3k"," if x = 2),(0","        "otherwise"):}`

1. Determine the value of k.
2. Find P(X < 2).
3. Find P(X > 2).

Answer 1

(i) It is known that the sum of probabilities of a probability distribution of random variables is one.

∴ k + 2k + 3k + 0 = 1

⇒ 6k = 1

k = `1/6`

(ii) P(X < 2) = P(X = 0) + P(X = 1)

= k + 2k

= 3k

`= 3/6`

`= 1/2`

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

= k + 2k + 3k

= 6k

= `6 xx 1/6`

= `6/6`

= 1

P(X ≥ 2) = P(X = 2) + P(X > 2)

= 3k + 0

= `3 xx 1/6`

= `3/6`

= `1/2`

(iii) P(X > 2) = 0.