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]] ...

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. E[E[X]]
C. E[X]
D. E[Y]

Accepted Answer

C. E[X] The Law of Total Expectation states that if you first compute the expected value of X for each fixed value of Y, and then average those conditional expectations over all values of Y, you recover the unconditional expectation of X. In symbols, E[E[X | Y]] = E[X]. This is a fundamental identity in probability theory used throughout statistics and machine learning.

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. E[E[X]] C. E[X] D. E[Y]