Question

Let \( X \) and \( Y \) be i.i.d. \( U(0,1) \) random variables. Then \( E(X|X>Y) \) equals

• A. \( \frac{1}{3} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{3}{4} \)

Hint:

For conditional expectations involving continuous random variables, use the formula \( E(X|X>Y) = \frac{\int_{y=0}^x x f_X(x) f_Y(y)}{P(X>Y)} \).

Answer 1

The Correct Option is C

Step 1: Define the conditional expectation.
We are asked to find \( E(X|X>Y) \), the conditional expectation of \( X \) given that \( X>Y \), where both \( X \) and \( Y \) are independent and uniformly distributed on [0,1]. Step 2: Set up the integral for the conditional expectation.
The conditional expectation is given by: \[ E(X|X>Y) = \frac{\int_0^1 \int_0^x x f_X(x) f_Y(y) \, dy \, dx}{P(X>Y)} \] where \( f_X(x) = 1 \) and \( f_Y(y) = 1 \) because \( X \) and \( Y \) are uniform random variables on [0,1]. Step 3: Solve the integral.
After solving the integral and simplifying, we find: \[ E(X|X>Y) = \frac{2}{3} \] Step 4: Conclusion.
Thus, the correct answer is \( \boxed{\frac{2}{3}} \).