Question

Consider that \( X \) and \( Y \) are independent continuous valued random variables with uniform PDF given by \( X \sim U(2, 3) \) and \( Y \sim U(1, 4) \). Then \( P(Y \leq X) \) is equal to \(\underline{\hspace{2cm}}\) (rounded off to two decimal places).

Hint:

For continuous uniform distributions, the probability of one variable being less than or equal to another can be found by integrating over the joint PDF.

Answer 1

Correct Answer: 0.45

The probability \( P(Y \leq X) \) can be computed by integrating the joint probability density function over the appropriate region: \[ P(Y \leq X) = \int_2^3 \int_1^x \frac{1}{(3-2)(4-1)} \, dy \, dx \] The joint probability density function is constant within the respective intervals. Simplifying the integral: \[ P(Y \leq X) = \frac{1}{3} \int_2^3 (x - 1) \, dx \] Evaluating the integral: \[ P(Y \leq X) = \frac{1}{3} \left[ \frac{(x - 1)^2}{2} \right]_2^3 = \frac{1}{3} \left[ \frac{1^2}{2} \right] = \frac{1}{6} \approx 0.17. \] Thus, \( P(Y \leq X) \approx 0.17 \).