# Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that

The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, Adam's law, and the smoothing theorem, among other names, states that if X is a random variable whose expected value E(X) is defined, and Y is any random variable on the same probability space, then

$$\begin{gather} E(X)=E(E(X|Y)) \end{gather}$$ i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X.

But for example Hayashi 's "Econometrics" states that the Law of total expectations is

$$\begin{gather} E(X)=E(E(X|Y)) \end{gather}$$

while, the Law of iterated expectations is

$$\begin{gather} E(X|Y)=E(E(X|Y,Z)|Y) \end{gather}$$ Is there an actual distinction between the two laws or are they one the general version of the other or something like this? If different, do they have different assumptions so that they are applicable in different contexts?

• I think the law of iterated expectations is a generalization of the law of total expectations, in your wording. But just as you knew(from the Wikipedia page) they are just the two names of the same thing. – Lerner Zhang Jun 4 '19 at 22:21

They are just two names for the same rule. The two particular cases you give in your question are analogous, but the latter is conditional on $$Y$$ and so it uses this as a conditioning variable on all statements. The latter equation is the most general form of the rule, since it allows you to specify a conditioning variable, but the former is also an application of the same rule.