Question

If X and Y are independent random variables with variances \(\sigma_x^2 = 5\) and \(\sigma_y^2 = 3\) then the variance of the random variable \(Z = 3X + 2Y - 5\) is

• A. 35
• B. 52
• C. 75
• D. 57

Hint:

For independent random variables, the variance of a linear combination \(Z = aX + bY + c\) is \(\operatorname{Var}(Z) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y)\).

Answer 1

The Correct Option is D

Concept:
For independent random variables, the variance of a linear combination \(Z = aX + bY + c\) is \(\operatorname{Var}(Z) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y)\). A constant added to the variable does not change the spread, so the constant term contributes nothing to the variance.

Step 1:
Identify the coefficients. Here \(a = 3\), \(b = 2\), and the constant is \(-5\). The variances are \(\operatorname{Var}(X) = 5\) and \(\operatorname{Var}(Y) = 3\).

Step 2:
Apply the formula and drop the constant term:
\[\operatorname{Var}(Z) = 3^2(5) + 2^2(3) = 9(5) + 4(3) = 45 + 12 = 57.\]

Answer: Option (4) — 57.