Question

Let \( X \) and \( Y \) be i.i.d. \( \text{Exp}(\lambda) \) random variables. If \( Z = \max\{X - Y, 0\} \), then which of the following statements is (are) TRUE?

• A. \( P(Z = 0) = \frac{1}{2} \)
• B. The cumulative distribution function of \( Z \) is \[ F(z) = \begin{cases} 0, & z < 0 \\ 1 - \frac{1}{2} e^{-\lambda z}, & z \geq 0 \end{cases} \]
• C. \( P(Z = 0) = 0 \)
• D. The cumulative distribution function of \( Z \) is \[ F(z) = \begin{cases} 0, & z < 0 \\ 1 - e^{-\lambda z}, & z \geq 0 \end{cases} \]

Hint:

For the maximum of two independent exponential random variables, the probability \( P(Z = 0) \) is \( \frac{1}{2} \).

Answer 1

The Correct Option is A, B

Step 1: Understand the distribution of \( Z = \max\{X - Y, 0\} \). 
Since \( X \) and \( Y \) are independent and exponentially distributed with parameter \( \lambda \), the probability that \( Z = 0 \) occurs when \( X \leq Y \), which happens with probability \( \frac{1}{2} \). Thus, \( P(Z = 0) = \frac{1}{2} \). 
Step 2: Analyzing the cumulative distribution function of \( Z \). 
The CDF of \( Z \), \( F(z) \), is given by: \[ F(z) = P(Z \leq z) = P(X - Y \leq z) = P(X \leq Y + z) \] For \( z \geq 0 \), we know: \[ F(z) = 1 - \frac{1}{2} e^{-\lambda z} \] This matches option (B). 
Step 3: Conclusion. 
The correct answer is A, as \( P(Z = 0) = \frac{1}{2} \).