Question

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x,y) = \begin{cases} \frac{1}{2} e^{-x}, & \text{if} \, |y| \leq x, \, x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Then \( E(X | Y = -1) \) equals

Hint:

When calculating conditional expectations, use the joint probability density function and integrate over the appropriate range to find the expected value.

Answer 1

Correct Answer: 2

Step 1: Understanding the problem.
We are given the joint probability density function \( f(x, y) \) for continuous random variables \( X \) and \( Y \). We need to calculate the conditional expectation \( E(X | Y = -1) \).
Step 2: Determining the conditional density.
The conditional probability density function \( f_{X | Y}(x | -1) \) is given by: \[ f_{X | Y}(x | -1) = \frac{f(x, -1)}{f_Y(-1)}. \] First, we compute \( f_Y(-1) \) by integrating the joint density over all possible values of \( x \): \[ f_Y(-1) = \int_1^\infty \frac{1}{2} e^{-x} \, dx = \frac{1}{2} e^{-1}. \] Thus, \( f_Y(-1) = \frac{1}{2} e^{-1} \).
Step 3: Computing the conditional expectation.
Now, using the conditional density, we compute \( E(X | Y = -1) \) as: \[ E(X | Y = -1) = \int_1^\infty x \cdot \frac{1}{2} e^{-x} \, dx. \] The integral evaluates to 2.0, so \( E(X | Y = -1) = 2.0 \).

Step 4: Conclusion.
Thus, \( E(X | Y = -1) = 2.0 \).