Show that for random variables X, Y and Z we have
E[E[E[x|y]|z]] = E[X]:
To show that E[E[E[X|Y]|Z]] = E[X], we need to use the properties of conditional expectation and the law of total expectation.
1. First, let's break down the expression step by step:
E[E[E[X|Y]|Z]]
2. According to the law of total expectation, we have:
E[E[X|Y]] = E[X]
3. Now, let's substitute this result into the original expression:
E[E[E[X|Y]|Z]] = E[E[X|Y|Z]]
4. Applying the law of total expectation once again, we get:
E[E[X|Y|Z]] = E[X|Z]
5. Finally, using the law of iterated expectations, we have:
E[X|Z] = E[X]
Therefore, E[E[E[X|Y]|Z]] = E[X].
This shows that the expected value of the conditional expectation of X given Y, when further conditioned on Z, is equal to the expected value of X.