Consider the following statements where X is a random variable and a, b are real numbers:
(A) \(E(aX+b)=aE(X)+b\)
(B) \(E(aX+b)=a^2E(X)+b\)
(C) \(\mathrm{Var}(aX+b)=a^2\,\mathrm{Var}(X)\)
(D) \(\mathrm{Var}(aX+b)=a\,\mathrm{Var}(X)+b\)
(E) \(\mathrm{Var}(X)=[E(X)]^2+E(X^2)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
Expectation is linear:
\(E(aX+b) = aE(X)+b\)
Adding b shifts every value and the mean by the same amount, so it does not change the spread; multiplying by a multiplies every deviation by a:
\(\mathrm{Var}(aX+b) = E\left[(aX-aE(X))^2\right] = a^2\,\mathrm{Var}(X)\)
\(\mathrm{Var}(X) = E(X^2)-[E(X)]^2\)
So \(E(aX+b) = aE(X)+b\), \(\mathrm{Var}(aX+b) = a^2\,\mathrm{Var}(X)\) and \(\mathrm{Var}(X) = E(X^2)-[E(X)]^2\).
Checking the statements
(A) \(E(aX+b)=aE(X)+b\) — Correct. Expectation is linear.
(B) \(E(aX+b)=a^2E(X)+b\) — Incorrect. The factor is a, not \(a^2\).
(C) \(\mathrm{Var}(aX+b)=a^2\,\mathrm{Var}(X)\) — Correct. The constant b does not change the spread, and a is squared.
(D) \(\mathrm{Var}(aX+b)=a\,\mathrm{Var}(X)+b\) — Incorrect. The variance is \(a^2\,\mathrm{Var}(X)\), with no b.
(E) \(\mathrm{Var}(X)=[E(X)]^2+E(X^2)\) — Incorrect. The square of the mean is subtracted: \(E(X^2)-[E(X)]^2\).