A random variable X has the following probability distribution:
| \(X\) | −1 | 0 | 1 |
| \(P(X)\) | 0.3 | 0.4 | 0.3 |
Which of the following statements are correct?
(A) \(E(X) = 0\)
(B) \(\mathrm{Var}(X) = 0.36\)
(C) \(E(2X + 3) = 3\)
(D) \(\mathrm{Var}(2X + 3) = 2.4\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
\(E(X) = (-1)(0.3) + 0(0.4) + 1(0.3) = 0\)
\(E(X^{2}) = 1(0.3) + 0(0.4) + 1(0.3) = 0.6\)
\(\mathrm{Var}(X) = 0.6 - 0^{2} = 0.6\)
For a linear function aX + b, \(E(aX + b) = aE(X) + b\) and \(\mathrm{Var}(aX + b) = a^{2}\,\mathrm{Var}(X)\):
\(E(2X + 3) = 2(0) + 3 = 3\)
\(\mathrm{Var}(2X + 3) = 4 \times 0.6 = 2.4\)
So \(E(X) = 0\), \(\mathrm{Var}(X) = 0.6\), \(E(2X + 3) = 3\) and \(\mathrm{Var}(2X + 3) = 2.4\).
Checking the statements
(A) \(E(X) = 0\) — Correct. The terms −0.3 and +0.3 cancel.
(B) \(\mathrm{Var}(X) = 0.36\) — Incorrect. \(\mathrm{Var}(X) = 0.6\); 0.36 is \(0.6^{2}\).
(C) \(E(2X + 3) = 3\) — Correct. \(2 \times 0 + 3 = 3\).
(D) \(\mathrm{Var}(2X + 3) = 2.4\) — Correct. \(2^{2} \times 0.6 = 2.4\).