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) $Var(aX+b)=a^2Var(X)$
(D) $Var(aX+b)=aVar(X)+b$
(E) $Var(X)=[E(X)]^2+E(X^2)$
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 2
$\text{Evaluate each statement:}$
$(A)\;E(aX+b)=aE(X)+b\;\text{— True}.$
$(B)\;E(aX+b)=a^2E(X)+b\;\text{— False}.$
$(C)\;\text{Var}(aX+b)=a^2\text{Var}(X)\;\text{— True}.$
$(D)\;\text{Var}(aX+b)=a\text{Var}(X)+b\;\text{— False}.$
$(E)\;\text{Var}(X)=[E(X)]^2+E(X^2)\;\text{— False, since }\text{Var}(X)=E(X^2)-[E(X)]^2.$
$\text{Correct statements are (A) and (C).}$