For a random variable X, \(E(X) = 2\) and \(\mathrm{Var}(X) = 3\).
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(E(X^{2})\) | (I) | 27 |
| (B) | \(E(3X - 1)\) | (II) | \(2\sqrt{3}\) |
| (C) | \(\mathrm{Var}(3X - 1)\) | (III) | 7 |
| (D) | Standard deviation of \(2X + 1\) | (IV) | 5 |
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) \(E(X^{2})\)
This matches (III) 7. From \(\mathrm{Var}(X) = E(X^{2}) - [E(X)]^{2}\):
\(E(X^{2}) = \mathrm{Var}(X) + [E(X)]^{2}\)
\(= 3 + 4 = 7\)
(B) \(E(3X - 1)\)
This matches (IV) 5. Since \(E(aX + b) = aE(X) + b\):
\(E(3X - 1) = 3(2) - 1 = 5\)
(C) \(\mathrm{Var}(3X - 1)\)
This matches (I) 27. Since \(\mathrm{Var}(aX + b) = a^{2}\,\mathrm{Var}(X)\):
\(\mathrm{Var}(3X - 1) = 9 \times 3 = 27\)
(D) Standard deviation of \(2X + 1\)
This matches (II) \(2\sqrt{3}\).
\(\mathrm{Var}(2X + 1) = 4 \times 3 = 12\)
\(\sigma = \sqrt{12} = 2\sqrt{3}\)