Question

If $Y$ is a random variable with variance 9, then Variance of $3Y - 9$ is

• A. 27
• B. 81
• C. 0
• D. 18

Hint:

Adding or subtracting a constant from a random variable does not change its dispersion. Thus, $\text{Var}(aY + b) = a^2 \text{Var}(Y)$ is completely independent of $b$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The variance of a linear transformation of a random variable is affected by scaling, but is unaffected by translation (shifts).
Key Formula or Approach:
For any random variable $Y$ and constants $a$ and $b$, the variance of $aY + b$ is given by:
\[ \text{Var}(aY + b) = a^2 \text{Var}(Y) \]

Step 2: Detailed Explanation:
We are given:
Variance of $Y$, $\text{Var}(Y) = 9$
We want to find $\text{Var}(3Y - 9)$.
Identify the parameters:
Scale factor, $a = 3$
Shift constant, $b = -9$
Substitute these parameters into the variance transformation formula:
\[ \text{Var}(3Y - 9) = 3^2 \text{Var}(Y) \]
\[ \text{Var}(3Y - 9) = 9 \times 9 = 81 \]
Therefore, the variance of $3Y - 9$ is 81.

Step 3: Final Answer
The correct option is (B).