The Use of Law of Iterated Expectations in this example.

Question

I am reading an solution to a problem, and I came across a expression:

$$E[X_{k+1}|X_{k}]=aX_{k}$$,

Where $$X_{k}$$ is the random variables of a series of trials.

The solution says that "by using the law of iterated expectations, we obtained

$$E[X_{k+1}]=aE[X_{k}]$$

So, I went to check what the Law of Iterated Expectation is, and I saw

$$E[E[X|Y]]=E[X]$$,

But I don't see how I can apply this law to

$$E[X_{k+1}|X_{k}]=aX_{k}$$,

How is the above derivation done? I don't see any connection....

Answer 1

Take expectation on both sides of $E(X_{k+1}|X_k)=aX_k$. You get $EX_{k+1}=EX_k$. You are just applying the Law of Iterated Expectation with $X=X_{k+1}$ and $Y=X_k$ and the fact that $EaX_k=aEX_k$.