Question

Let $X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$ be two normally distributed random variables, where $\mu_1 = 2, \mu_2 = 3$ and $\sigma_1^2 = 4, \sigma_2^2 = 9$. The correlation coefficient between them is 0.5. The variance of the random variable $(X_1 + X_2)$ is ___________. (in integer) 
 

Hint:

When two variables are correlated, the variance of their sum includes the covariance term: $\mathrm{Cov}(X_1, X_2) = \rho \sigma_1 \sigma_2$.

Answer 1

Correct Answer: 19

Step 1: Formula for variance of the sum of two correlated variables.
\[ \mathrm{Var}(X_1 + X_2) = \sigma_1^2 + \sigma_2^2 + 2\rho \sigma_1 \sigma_2. \]
Step 2: Substitute given values.
\[ \sigma_1^2 = 4, \ \sigma_2^2 = 9, \ \rho = 0.5, \ \sigma_1 = 2, \ \sigma_2 = 3. \] \[ \mathrm{Var}(X_1 + X_2) = 4 + 9 + 2(0.5)(2)(3) = 13 + 6 = 19. \]
Step 3: Conclusion.
\[ \boxed{\mathrm{Var}(X_1 + X_2) = 19.} \]