If a random variable X has the following probability distribution:
| X | 0 | 1 | 2 | 3 |
| P(X) | \(K\) | \(\displaystyle\frac{K}{2}\) | \(\displaystyle\frac{K}{4}\) | \(\displaystyle\frac{K}{8}\) |
then,
Match List-I with List-II
| List-I | List-II | ||
| (A) | The value of \(K\) is | (I) | 2/15 |
| (B) | \(P(0 \lt X \lt 2)\) is | (II) | 1/15 |
| (C) | \(P(1 \lt X \lt 3)\) is | (III) | 8/15 |
| (D) | \(P(X \gt 2)\) is | (IV) | 4/15 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
(A) The value of \(K\) is
This matches (III) 8/15. The sum of all probabilities is 1:
\( \displaystyle K + \frac{K}{2} + \frac{K}{4} + \frac{K}{8} = 1 \)
\( \displaystyle \frac{15K}{8} = 1,\quad K = \frac{8}{15} \)
(B) \(P(0 \lt X \lt 2)\) is
This matches (IV) 4/15. The only value strictly between 0 and 2 is X = 1:
\( \displaystyle P(X = 1) = \frac{K}{2} = \frac{4}{15} \)
(C) \(P(1 \lt X \lt 3)\) is
This matches (I) 2/15. The only value strictly between 1 and 3 is X = 2:
\( \displaystyle P(X = 2) = \frac{K}{4} = \frac{2}{15} \)
(D) \(P(X \gt 2)\) is
This matches (II) 1/15. The only value greater than 2 is X = 3:
\( \displaystyle P(X = 3) = \frac{K}{8} = \frac{1}{15} \)
So the matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II).