If X is a random variable and a, b are real numbers, then which of the following statements are correct?
(A) \(E[aX+b] = a\,E(X) + b\)
(B) \(\mathrm{Var}\,(aX + b) = a^2\,\mathrm{Var}\,(X) + b\)
(C) \(\mathrm{Var}\,(aX + b) = a\,\mathrm{Var}\,(X)\)
(D) \(\mathrm{Var}\,(X) = E(X^2) - [E(X)]^2\)
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
For constants a and b, the expectation and variance of aX + b are
\(E(aX + b) = a\,E(X) + b\)
\(\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)\)
Adding b shifts every value by the same amount, so the spread does not change. Multiplying by a multiplies every deviation from the mean by a, so the variance is multiplied by \(a^2\).
So \(E(aX + b) = a\,E(X) + b\) and \(\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)\).
Checking the statements
(A) \(E[aX+b] = a\,E(X) + b\) — Correct. The mean is scaled by a and shifted by b.
(B) \(\mathrm{Var}\,(aX + b) = a^2\,\mathrm{Var}\,(X) + b\) — Incorrect. \(\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)\), with no b.
(C) \(\mathrm{Var}\,(aX + b) = a\,\mathrm{Var}\,(X)\) — Incorrect. The factor is \(a^2\), not a.
(D) \(\mathrm{Var}\,(X) = E(X^2) - [E(X)]^2\) — Correct. With \(\mu = E(X)\):
\(\mathrm{Var}(X) = E[(X - \mu)^2]\)
\(= E(X^2) - 2\mu E(X) + \mu^2\)
\(= E(X^2) - [E(X)]^2\)