QuestionProblem 33 Let $X$ and $Y$ be two random variables. Suppose that $\sigma_{X}^{2}=4$, and $\sigma_{Y}^{2}=9$. If we know that the two random variables $Z=2 X-Y$ and $W=X+Y$ are independent, find $\operatorname{Cov}(X, Y)$ and $\rho(X, Y)$.

Studdy Solution
Calculate $\rho(X, Y)$ using $\operatorname{Cov}(X, Y)$:
$\rho(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}$
$= \frac{1}{\sqrt{4} \times \sqrt{9}}$
$= \frac{1}{2 \times 3}$
$= \frac{1}{6}$
The covariance and correlation coefficient are:
$\operatorname{Cov}(X, Y) = 1$
$\rho(X, Y) = \frac{1}{6}$