Question

Let \( X \) and \( Y \) have the joint probability density function \[ f(x, y) = \begin{cases} e^{-y}, & 0 < x < y < \infty, \\ 0, & \text{otherwise}. \end{cases} \] Then the correlation coefficient between \( X \) and \( Y \) equals 
 

• A. \( \frac{1}{3} \)
• B. \( \frac{1}{\sqrt{3}} \)
• C. \( \frac{1}{\sqrt{2}} \)
• D. \( \frac{2}{\sqrt{3}} \)

Hint:

When computing the correlation coefficient, first calculate the marginal distributions, then compute the covariance and variances of the variables.

Answer 1

The Correct Option is C

Step 1: Compute the expected values. 
The joint probability density function \( f(x, y) \) is given. To compute the correlation coefficient, we first need to compute the marginal distributions of \( X \) and \( Y \), and then their expected values \( E(X) \), \( E(Y) \), and \( E(XY) \).

Step 2: Calculate the variances and covariance. 
Using the definitions of variance and covariance, compute the necessary moments from the joint distribution. Finally, the correlation coefficient is given by: \[ \rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}. \]

Step 3: Conclusion. 
The correct answer is (C) \( \frac{1}{\sqrt{2}} \).