I am studying a proof of a theorem in stohastic processes and there is this part thas confuses me :
$E[T_A] = \sum_{y \in X} P[X_1=y|Xo=x]E[T_A|X_0 =x, X_1 =y]$
where $T_A = inf\{k \geq 0: X_k \in A\}$
I do not get it because how come we have into the sum $E[T_A]$?
Since, $E[X] = \sum px$ where x is every possible value of random variable $X$.
So , we are trying to say something like :
The fact that Markov chain $\{X_n\}_{n \ in No}$ is hitting $A$
is the same with the fact $\{X_0=1, X_1=y\}\wedge \{X_n\in A \}$ for all possible $(x,y)$ - of course , that's why we have a sum - . And because we are interested in hitting time ... in a similar way we can say that the fact of $\{X_n\}_{n \ in No}$ is hitting $A$ for the first time is equivalent to :
$\{X_0=1, X_1=y\}\wedge \{T_A =n\}$ .
So it would make more sense to me to write something like:
for $T_A \in N_0 $
$E[T_A] = \sum_{y \in X} \sum_{n\in N_0}n P[T_A=n|Xo=x,X_1 = y]$
