The probability distribution of a random variable X is given below:

(i) W.k.t P(0) + P(1) + P(2) + P(3) = 1

⇒ k + k/2 + k/4 + k/8 = 1

(8k + 4k + 2k + k)/8 = 1

15k = 8

Hence, k = 8/15

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

= k + k/2 + k/4 = 7k/4 = 7/4 x 8/15 = 14/15

And P(X > 2) = P(X = 3) = k/8 = 1/8 x 8/15 = 1/15

(iii) P(X ≤ 2) + P(X ≥ 2) = 14/15 + 1/15 = (14 + 1)/15 = 15/15 = 1