We are asked to compute $\text{Var}(X + 2Y)$ given the following information:

• $\text{Corr}(X, Y) = \frac{1}{221}$
• $\text{Var}(X) = 1$
• $\text{Var}(Y) = 2$

Step 1: Formula for Variance of a Linear Combination

For two random variables $X$ and $Y$, the variance of a linear combination $aX + bY$ is given by: $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(X, Y)$ In our case, we want to compute $\text{Var}(X + 2Y)$. Here:

• $a = 1$ (coefficient of $X$)
• $b = 2$ (coefficient of $Y$)

So, the formula becomes: $\text{Var}(X + 2Y) = 1^2 \text{Var}(X) + 2^2 \text{Var}(Y) + 2 \cdot 1 \cdot 2 \cdot \text{Cov}(X, Y)$

Step 2: Use Given Values

We have $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and we know the correlation coefficient between $X$ and $Y$ is: $\text{Corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}$ Substituting the known values: $\frac{1}{221} = \frac{\text{Cov}(X, Y)}{\sqrt{1 \cdot 2}} = \frac{\text{Cov}(X, Y)}{\sqrt{2}}$ Thus, solving for $\text{Cov}(X, Y)$: $\text{Cov}(X, Y) = \frac{\sqrt{2}}{221}$

Step 3: Calculate $\text{Var}(X + 2Y)$

Substituting all values into the variance formula: $\text{Var}(X + 2Y) = 1^2 \cdot 1 + 2^2 \cdot 2 + 2 \cdot 1 \cdot 2 \cdot \frac{\sqrt{2}}{221}$ $\text{Var}(X + 2Y) = 1 + 8 + 4 \cdot \frac{\sqrt{2}}{221}$ Approximating $\frac{\sqrt{2}}{221} \approx 0.0064$: $\text{Var}(X + 2Y) = 1 + 8 + 4 \cdot 0.0064 = 9 + 0.0256 = 9.0256$

Thus, the variance of $X + 2Y$ correct to three decimal places is: $\boxed{9.026}$

Would you like more details, or do you have any other questions?

Here are 5 related questions for practice:

1. How does the correlation coefficient affect the covariance between two variables?
2. How would the result change if $\text{Var}(Y)$ were increased to 3?
3. What happens to the variance if $X$ and $Y$ are independent (i.e., $\text{Corr}(X, Y) = 0$)?
4. How do we compute the covariance if the correlation coefficient is negative?
5. What is the formula for the variance of a sum of three random variables?

Tip: The correlation coefficient is always between -1 and 1 and indicates the strength and direction of a linear relationship between two variables.