If X is a random variable and a, b are real numbers, then which of the following statements are correct?
(A) $\text{E[aX+b] = a E(X) + b}$
(B) $\text{Var (ax + b)} = a^2 \text{Var (X) + b}$
(C) $\text{Var (ax + b) = a Var (X)}$
(D) $\text{Var (X)} = E(X^2) - [E(X)]^2$
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
Given properties of random variable $X$ and real numbers $a, b$:
(A) $E[aX+b] = a E(X) + b$ → True (linearity of expectation)
(B) $\text{Var}(aX+b) = a^2 \text{Var}(X) + b$ → False, correct formula: $\text{Var}(aX+b) = a^2 \text{Var}(X)$
(C) $\text{Var}(aX+b) = a \text{Var}(X)$ → False, factor should be $a^2$
(D) $\text{Var}(X) = E(X^2) - [E(X)]^2$ → True (definition of variance)