Question

Let \( X \) and \( Y \) be two random variables with mean 0, variance 1, and correlation coefficient \( \frac{1}{3} \). Then the value of \( {Var}(X + 3Y) \) is equal to:

• A.
• B.
• C.
• D.

Hint:

To calculate the variance of a linear combination of random variables, use the formula: \[ {Var}(aX + bY) = a^2 {Var}(X) + b^2 {Var}(Y) + 2ab \, {Cov}(X, Y) \]

Answer 1

The Correct Option is D

We are given that \( X \) and \( Y \) are random variables with the following properties:
- \( \mu_X = 0 \), \( \mu_Y = 0 \) (mean of both variables is 0),
- \( {Var}(X) = 1 \), \( {Var}(Y) = 1 \) (variance of both variables is 1),
- The correlation coefficient between \( X \) and \( Y \) is \( \rho = \frac{1}{3} \).
We need to calculate \( {Var}(X + 3Y) \).
The formula for the variance of the sum of two random variables is:
\[ {Var}(X + 3Y) = {Var}(X) + 9 \cdot {Var}(Y) + 2 \cdot 3 \cdot {Cov}(X, Y) \] Given that \( {Var}(X) = 1 \), \( {Var}(Y) = 1 \), and the correlation coefficient \( \rho = \frac{1}{3} \), the covariance \( {Cov}(X, Y) \) is given by:
\[ {Cov}(X, Y) = \rho \cdot \sigma_X \cdot \sigma_Y = \frac{1}{3} \cdot 1 \cdot 1 = \frac{1}{3} \] Now, substituting the values into the variance formula:
\[ {Var}(X + 3Y) = 1 + 9 \cdot 1 + 2 \cdot 3 \cdot \frac{1}{3} \] Simplifying:
\[ {Var}(X + 3Y) = 1 + 9 + 2 = 12 \] Thus, the value of \( {Var}(X + 3Y) \) is 12.