Question

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} 8xy, & 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases} \] Then \( 9 \, \text{Cov}(X, Y) \) equals

Hint:

When calculating covariance for continuous random variables, remember to first compute the expected values and then use the covariance formula \( \text{Cov}(X, Y) = E[XY] - E[X]E[Y] \).

Answer 1

Correct Answer: 0.15 - 0.17

Step 1: Understanding the problem.
We are given the joint probability density function \( f(x, y) \) of continuous random variables \( X \) and \( Y \). To calculate the covariance \( \text{Cov}(X, Y) \), we use the formula: \[ \text{Cov}(X, Y) = E[XY] - E[X]E[Y]. \]
Step 2: Finding \( E[X] \) and \( E[Y] \).
We first need to calculate the expected values \( E[X] \) and \( E[Y] \). These are calculated by integrating the joint density function: \[ E[X] = \int_0^1 \int_0^x x \cdot f(x, y) \, dy \, dx. \] Similarly, \[ E[Y] = \int_0^1 \int_0^x y \cdot f(x, y) \, dy \, dx. \]
Step 3: Finding \( E[XY] \).
Next, we calculate \( E[XY] \) using the following integral: \[ E[XY] = \int_0^1 \int_0^x xy \cdot f(x, y) \, dy \, dx. \]
Step 4: Calculating Covariance.
Now, substitute the values of \( E[X] \), \( E[Y] \), and \( E[XY] \) into the covariance formula. After evaluating the integrals, we find that \( 9 \, \text{Cov}(X, Y) \) is approximately between 0.15 and 0.17.

Step 5: Conclusion.
Thus, the value of \( 9 \, \text{Cov}(X, Y) \) is approximately between 0.15 and 0.17.