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
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]
The correct statement is:
c) E[E[X/Y,Z]] = E[X/Z]
Why?
To understand why option c) is the correct statement, let's break down the equation step by step.
E[E[X/Y,Z]] can be interpreted as the expected value of X given Y and Z, where the outer expectation is taken over Y and the inner expectation is taken over X given Y and Z. This implies that Y and Z are treated as fixed values when computing the inner expectation.
On the other hand, E[X/Z] is the expected value of X given Z. In this case, Z is treated as a fixed value when computing the expectation.
Now, the law of iterated expectations states that E[E[X/Y]] = E[X]. In our case, if we apply this law in a conditional universe given Z, it tells us that E[E[X/Y,Z]] = E[X/Z]. This means that we can express the expected value of X given Y and Z as the expected value of X given Z.
Therefore, option c) E[E[X/Y,Z]] = E[X/Z] is true.
The correct statement is (b) E[E[X/Y]/Z] = E[X/Z].
The law of iterated expectations states that for any two random variables X and Y, we have E[E[X/Y]] = E[X].
When we want to apply this law in a conditional universe, given another random variable Z, we need to take into account that the conditioning now includes Z.
So, E[X/Z] represents the expectation of X, given Z. In this case, the conditioning only involves Z.
On the other hand, when we evaluate E[E[X/Y]/Z], it means we are first taking the conditional expectation of X given Y and then taking the expectation of that result, given Z. In this case, both Y and Z are part of the conditioning.
Therefore, option (b) E[E[X/Y]/Z] = E[X/Z] is the correct statement.
To determine if the statements are true or false, let's first break down the law of iterated expectations and understand what it really means.
The law of iterated expectations states that the unconditional expectation of a conditional expectation is equal to the unconditional expectation of the original variable. In other words, if we have a random variable X and two other random variables Y and Z, we can calculate the expectation of X in two steps:
1. First, we calculate the conditional expectation of X given Y: E[X/Y].
2. Then, we take the unconditional expectation of the conditional expectation calculated in step 1: E[E[X/Y]].
Now, let's analyze the statements:
a) E[E[X/Y,Z]/Z] = E[X/Z]
This statement is false. In this statement, we are calculating the conditional expectation of X given both Y and Z, and then taking the expectation with respect to Z. The law of iterated expectations does not apply in this case. The correct result should be E[E[X/Y,Z]] = E[X/Y].
b) E[E[X/Y]/Z] = E[X/Z]
This statement is also false. In this statement, we are calculating the conditional expectation of X given Y and then taking the expectation with respect to Z. Once again, the law of iterated expectations does not apply in this case. The correct result should be E[E[X/Y]/Z] = E[X/Y].
c) E[E[X/Y,Z]] = E[X/Z]
This statement is true. In this statement, we are calculating the conditional expectation of X given both Y and Z. The law of iterated expectations applies here, and the result is indeed equal to E[X/Z].
So, the correct answer is:
c) E[E[X/Y,Z]] = E[X/Z]