Suppose X*X* and Y*Y* are two random variables such that correlation coefficient between X*X* and Y*Y* is 1221​, Var(X)=1*Var*(*X*)=1 and Var(Y)=2.*Var*(*Y*)=2. Compute Var(X+2Y)*Var*(*X*+2*Y*), correct to three decimal places.




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.

Suppose X*X* and Y*Y* are two random variables such that correlation coefficient between X*X* and Y*Y* is 1221​, Var(X)=1*Var*(*X*)=1 and Var(Y)=2.*Var*(*Y*)=2. Compute Var(X+2Y)*Var*(*X*+2*Y*), correct to three decimal places.

Learn how to compute the variance of X + 2Y given Var(X) = 1, Var(Y) = 2, and Corr(X, Y) = 1/221. This problem involves key statistical concepts such as correlation, covariance, and variance, suitable for undergraduate-level probability and statistics courses.

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