題組內容

I. Let the joint pdf of random variables X and Y be given by $$f(x, y) = \begin{cases} cx & 0 < x < y < 1 \\ 0 & \text{elsewhere.} \end{cases}$$

5. (5%) Suppose you are dealing with an unknown joint pdf $ f(x, y) $. Suppose you only have two random samples of size $ n $: one is $ X_1, X_2, \ldots, X_n $ from the distribution of $ X $, and the other is $ W_1, W_2, \ldots, W_n $ from the distribution of $ W $, where $ W = E(X|Z) $. Can you determine which random sample would allow you to derive a more precise estimator of $ E(X) $? Provide an explanation for your choice.