Question

Suppose \( X \) and \( Y \) are random variables. The conditional expectation of \( X \) given \( Y \) is denoted by \( E[X | Y] \). Then \( E[E[X | Y]] \) equals:

• A. \( E[X | Y] \)
• B. \( \frac{E[X]}{E[Y]} \)
• C. \( E[X] \)
• D. \( E[Y] \)

Hint:

The law of iterated expectations (also called the tower rule) is very useful when simplifying problems involving conditional expectations. It tells us that the expectation of the conditional expectation of a random variable equals the expectation of the random variable itself.

Answer 1

The Correct Option is C

The problem asks about \( E[E[X | Y]] \), where \( E[X | Y] \) is the conditional expectation of \( X \) given \( Y \). By the law of iterated expectations (also known as the tower rule), we have the following relationship: \[ E[E[X | Y]] = E[X]. \] This rule states that the expectation of the conditional expectation of \( X \) given \( Y \) is equal to the overall expectation of \( X \). 
This holds because the inner expectation \( E[X | Y] \) is itself a random variable that is a function of \( Y \), and taking the expectation over \( Y \) gives the total expectation of \( X \). 
Thus, the correct answer is \( E[X] \), which corresponds to option (C).