Another question. The law of iterated expectations tells us that E[E[X/Y]] = E[X]. Suppose that we want apply this law in a conditional universe, given another random variable Z, in order to evaluate E[X/Z]. Then, tell me if this is true or false, remember than more than one option can be true
a) E[E[X/Y,Z]/Z] = E[X/Z]
b) E[E[X/Y]/Z] = E[X/Z]
c) E[E[X/Y,Z]] = E[X/Z]
a) True
b) False
c) False
why?
The correct answer is:
a) E[E[X/Y,Z]/Z] = E[X/Z]
This is true because according to the law of iterated expectations, we can simplify the expression E[E[X/Y,Z]/Z] as E[X/Y,Z]. Then, when we take the expectation of this expression with respect to Z, it collapses to E[X/Z]. Therefore, a) is true.
b) E[E[X/Y]/Z] = E[X/Z]
This is false because the law of iterated expectations does not allow us to swap the order of conditioning. In this case, E[E[X/Y]/Z] means taking the expectation of X conditional on Y first, and then taking the expectation of that result conditional on Z. But E[X/Z] means taking the expectation of X conditional on Z. Since these two are different conditioning sequences, b) is false.
c) E[E[X/Y,Z]] = E[X/Z]
This is also false for the same reason as option b). The law of iterated expectations does not allow us to swap the order of conditioning. E[E[X/Y,Z]] means taking the expectation of X conditional on Y and Z, while E[X/Z] means taking the expectation of X conditional on Z. Since the order of conditioning is different, c) is false.
To determine if the given statements are true or false, we'll evaluate them one by one:
a) E[E[X/Y,Z]/Z] = E[X/Z]
This statement is false. The law of iterated expectations does not allow for dividing the inner expectation by the conditioning variable. The covariance of Y and Z may not be zero, resulting in a situation where dividing by Z inside the conditional expectation is not valid.
b) E[E[X/Y]/Z] = E[X/Z]
This statement is true. According to the law of iterated expectations, we can separate the conditional expectations. Therefore, E[E[X/Y]/Z] is equal to E[X/Z].
c) E[E[X/Y,Z]] = E[X/Z]
This statement is true. The law of iterated expectations allows for the inner expectation to be conditioned on multiple random variables. Therefore, E[E[X/Y,Z]] is equal to E[X/Z].
In summary:
a) False
b) True
c) True