Question

If variance of \(X\) is \(4\), then the variance of \(Y=4X+2\) is

• A. \(64\)
• B. \(16\)
• C. \(4\)
• D. \(0\)

Hint:

For \[ Y=aX+b, \] \[ \boxed{\operatorname{Var}(Y)=a^2\operatorname{Var}(X).} \] Adding a constant does not affect variance.

Answer 1

The Correct Option is A

Step 1: Go back to the definition of variance.
$\operatorname{Var}(Y)=E\big[(Y-E(Y))^2\big]$. With $Y=4X+2$, we have $E(Y)=4E(X)+2$, so $Y-E(Y)=4X+2-(4E(X)+2)=4\big(X-E(X)\big)$.
Step 2: Square this difference.
$(Y-E(Y))^2=16\big(X-E(X)\big)^2$, and taking expectation on both sides gives $\operatorname{Var}(Y)=16\operatorname{Var}(X)$.
Step 3: Plug in the given variance.
$\operatorname{Var}(X)=4$, so $\operatorname{Var}(Y)=16\times4=64$.
\[ \boxed{64} \]