Question

Let $X$ and $Y$ be continuous random variables having the joint probability density function
$f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\\0, & \text{otherwise}\end{cases}$
Then which of the following statements is/are correct?

• A. $P(Y^2 = 3X) = 0$
• B. $P(X > 2Y) = \frac{1}{2}$
• C. $P(X - Y \geq 1) = e^{-1}$
• D. $P(X > \ln 2 \mid Y > \ln 3) = 0$

Answer 1

Answer: (A), (B), (C)

$P(Y^2 = 3X) = 0$

Answer 2

The correct option is (A): $P(Y^2 = 3X) = 0$,(B):,$P(X > 2Y) = \frac{1}{2}$,(C): $P(X - Y \geq 1) = e^{-1}$