I am studying stohastic processes, currently crossroad processes so I have to prove that: $$VARX_{n}=E(VAR(X_{n}|X_{n-1}))+VAR(E(X_{n}|X_{n-1}))$$ Where $X_{n}$ is the current position and $X_{n-1}$ is the previous.
We are given that:
$$EX_{n}=E(E(X_{n}|X_{n-1}))$$ and Wald's equation: $$EZ=EXEM,Z=\sum_{i=1}^{M}x_{i}$$ My attempt: $$VarX_{n}=E(E(X_{n}|X_{n-1}))^{2}-E(E(X_{n}|X_{n-1})^{2})$$ $$VarX_{n}=E(E(X_{n}|X_{n-1}))^{2}-E(\sum x_{i} P(X_{n}|X_{n-1}))^{2}$$ $$VarX_{n}=E(E(X_{n}|X_{n-1}))^{2}-E(\frac{\sum x_{i}P(X_{n}X_{n-1})}{P(X_{n-1})})^{2}$$
And here I realise that this is not the way ..
Can you please show me the proof of this statement or suggest me a resource where I can find the answer?
This comes from a more general result, which is:
$$Var(X)=E(Var(X|Y))+Var(E(X|Y))$$
See e.g., formula *** in (https://www.ma.utexas.edu/users/mks/384G04/condmeanvar3.pdf)