A random variable X takes the values 0, 1 and 2. Which of the following can be the probability distribution of X?
(A) \(P(X = 0) = 0.2,\ P(X = 1) = 0.5,\ P(X = 2) = 0.3\)
(B) \(P(X = 0) = 0.4,\ P(X = 1) = 0.4,\ P(X = 2) = 0.4\)
(C) \(P(X = 0) = 0.6,\ P(X = 1) = -0.1,\ P(X = 2) = 0.5\)
(D) \(P(X = 0) = \displaystyle\frac{1}{4},\ P(X = 1) = \frac{1}{2},\ P(X = 2) = \frac{1}{4}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
A list of probabilities is a probability distribution only when every probability lies between 0 and 1 and all the probabilities add up to 1.
So each set must have no negative value and a total of exactly 1.
Checking the statements
(A) 0.2, 0.5, 0.3 — Correct. All are non-negative and 0.2 + 0.5 + 0.3 = 1.
(B) 0.4, 0.4, 0.4 — Incorrect. The total is 1.2, not 1.
(C) 0.6, −0.1, 0.5 — Incorrect. The total is 1, but −0.1 is negative and a probability cannot be negative.
(D) \(\frac{1}{4}, \frac{1}{2}, \frac{1}{4}\) — Correct. All are non-negative and \(\frac{1}{4} + \frac{1}{2} + \frac{1}{4} = 1\).