Question

Two independent random variables X and Y have variances 0.2 and 0.5 respectively. Let Z = 5X - 2Y. The variance of Z is

• A. 3
• B. 7
• C. 4
• D. 5

Hint:

Remember that the variance of a difference is not the difference of the variances.
Because coefficients are squared, we have $\text{Var}(aX - bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$.
The negative sign becomes positive.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the variance of a new random variable $Z$ which is defined as a linear combination of two independent random variables $X$ and $Y$ with known variances.

Step 2: Key Formula or Approach:
For any two independent random variables $X$ and $Y$, the variance of their linear combination is given by:
\[ \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) \] Where $a$ and $b$ are real constant coefficients.

Step 3: Detailed Explanation:

• List the given values:
\[ \text{Var}(X) = 0.2 \] \[ \text{Var}(Y) = 0.5 \]
• The linear combination is defined as:
\[ Z = 5X - 2Y \]
• Identify the coefficients:
\[ a = 5, \quad b = -2 \]
• Apply the variance formula:
\[ \text{Var}(Z) = (5)^2 \text{Var}(X) + (-2)^2 \text{Var}(Y) \] \[ \text{Var}(Z) = 25 \text{Var}(X) + 4 \text{Var}(Y) \]
• Substitute the known values of the variances:
\[ \text{Var}(Z) = 25(0.2) + 4(0.5) \]
• Calculate each term:
\[ 25 \times 0.2 = 5 \] \[ 4 \times 0.5 = 2 \]
• Sum the results:
\[ \text{Var}(Z) = 5 + 2 = 7 \]

Step 4: Final Answer:
The variance of $Z$ is 7, which corresponds to option (B).