A random variable X has the following probability distribution:
| \(X\) | −2 | −1 | 0 | 1 | 2 | 3 |
| \(P(X)\) | 0.1 | 0.2 | \(k\) | 0.3 | \(2k\) | 0.1 |
then which of the following are TRUE?
(A) \(k = 0.1\)
(B) \(P(X \lt 1) = 0.4\)
(C) \(P(X \lt 2) = 0.7\)
(D) \(P(0 \lt x \lt 3) = 0.5\)
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B), (C) and (D)
The probabilities add up to 1:
\( \displaystyle 0.1 + 0.2 + k + 0.3 + 2k + 0.1 = 1 \)
\( \displaystyle 0.7 + 3k = 1 \)
\( \displaystyle k = 0.1 \)
So k = 0.1 and \(P(-2) = 0.1,\ P(-1) = 0.2,\ P(0) = 0.1,\ P(1) = 0.3,\ P(2) = 0.2,\ P(3) = 0.1\).
Checking the statements
(A) \(k = 0.1\) — Correct. \(0.7 + 3k = 1\) gives \(k = 0.1\).
(B) \(P(X \lt 1) = 0.4\) — Correct. X = −2, −1, 0: \(0.1 + 0.2 + 0.1 = 0.4\).
(C) \(P(X \lt 2) = 0.7\) — Correct. X = −2, −1, 0, 1: \(0.1 + 0.2 + 0.1 + 0.3 = 0.7\).
(D) \(P(0 \lt x \lt 3) = 0.5\) — Correct. X = 1, 2 (strictly between 0 and 3): \(0.3 + 0.2 = 0.5\).