The correct statement is:
c) E[E[X/Y,Z]] = E[X/Z]
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]
3 answers
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.
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.
3 answers