Question

The random variable \(Y \sim U(0, X)\), where the marginal density of \(X\) isv

\( f_X(x) = \begin{cases} 2x & \text{for } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases} \)

Then \(E(Y)\) is:

• A. $2/3$
• B. $4/3$
• C. $1/2$
• D. $1/3$

Hint:

The Law of Iterated Expectations ($E(Y) = E[E(Y|X)]$) is much faster than finding the joint density $f(x,y)$ and performing a double integral. Always look for hierarchical structures in probability problems.

Answer 1

The Correct Option is D