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

Step 1: Recall the rule for variance of a sum.
For independent variables, variance of \(aX+bY+c\) only uses the squares of the coefficients on the random parts.

Step 2: Note the constant drops out.
The \(-5\) just shifts \(Z\); it does not change the spread, so it adds nothing to the variance.

Step 3: Square the coefficient of X.
Coefficient of \(X\) is 3, so \(3^2=9\), and \(9\times\sigma_x^2=9\times5=45\).

Step 4: Square the coefficient of Y.
Coefficient of \(Y\) is 2, so \(2^2=4\), and \(4\times\sigma_y^2=4\times3=12\).

Step 5: Add the two contributions.
\[\operatorname{Var}(Z)=45+12=57.\]
Step 6: Pick the option.
This is option 4.
\[ \boxed{\operatorname{Var}(Z)=57} \]