Variance Calculation for Random Variables

We are given two random variables, $X$ and $Y$, with the following statistical properties:

• Means: $E[X] = 0$, $E[Y] = 0$
• Variances: $\text{Var}(X) = 1$, $\text{Var}(Y) = 1$
• Correlation Coefficient: $\rho_{XY} = \frac{1}{3}$

The objective is to find the value of $\text{Var}(X + 3Y)$.

Variance Formula for Linear Combinations

The general formula for the variance of a linear combination of two random variables, $aX + bY$, is:

$ \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(X, Y) $

In this problem, $a=1$ and $b=3$. We first need to find the covariance $\text{Cov}(X, Y)$.

Covariance Calculation

The covariance is related to the correlation coefficient by the formula:

$ \text{Cov}(X, Y) = \rho_{XY} \sqrt{\text{Var}(X)} \sqrt{\text{Var}(Y)} $

Substituting the given values:

$ \text{Cov}(X, Y) = \frac{1}{3} \sqrt{1} \sqrt{1} = \frac{1}{3} $

Final Variance Calculation

Now, we apply the variance formula with $a=1$, $b=3$, $\text{Var}(X)=1$, $\text{Var}(Y)=1$, and $\text{Cov}(X, Y)=\frac{1}{3}$:

$ \text{Var}(X + 3Y) = (1)^2 \text{Var}(X) + (3)^2 \text{Var}(Y) + 2(1)(3) \text{Cov}(X, Y) $

$ \text{Var}(X + 3Y) = 1 \cdot (1) + 9 \cdot (1) + 6 \cdot \left(\frac{1}{3}\right) $

$ \text{Var}(X + 3Y) = 1 + 9 + 2 $

$ \text{Var}(X + 3Y) = 12 $

The value of $\text{Var}(X + 3Y)$ is 12.