If X is a random variable and a, b are real numbers, then which of the following statements are true?
(A) \(\mathrm{Var}(aX+b) = a^2\, \mathrm{Var}(X)\)
(B) \(E(aX+b)= a\,E(X) + b\)
(C) \(E(aX+b)= a\,E(X) - E(b)\)
(D) \(\mathrm{Var}(aX+b)= a\,\mathrm{Var}(X) + b\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
By the linearity of expectation, and since the mean of a constant is the constant itself (\(E(b) = b\)):
\(E(aX + b) = a\,E(X) + b\)
Adding b shifts every value and the mean by b, so the deviations from the mean do not change; multiplying by a multiplies every deviation by a, so the squared deviations are multiplied by \(a^2\):
\(\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)\)
So \(E(aX + b) = a\,E(X) + b\) and \(\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)\).
Checking the statements
(A) \(\mathrm{Var}(aX+b) = a^2\, \mathrm{Var}(X)\) — Correct. This is the variance rule.
(B) \(E(aX+b)= a\,E(X) + b\) — Correct. This is the expectation rule.
(C) \(E(aX+b)= a\,E(X) - E(b)\) — Incorrect. \(E(b) = b\), so the right side is \(a\,E(X) - b\), not \(a\,E(X) + b\).
(D) \(\mathrm{Var}(aX+b)= a\,\mathrm{Var}(X) + b\) — Incorrect. The variance is \(a^2\,\mathrm{Var}(X)\); b does not affect it.