# Markov Chain Conditional Expectation

• Nov 23rd 2010, 06:36 AM
tunaaa
Markov Chain Conditional Expectation
Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from http://latex.codecogs.com/gif.latex?...5Comega%20%29], along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks
• Nov 27th 2010, 01:32 PM
Focus
Quote:

Originally Posted by tunaaa
Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from http://latex.codecogs.com/gif.latex?...5Comega%20%29], along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks

The first two are equal by definition. The actual meaning of conditioning is introducing a new measure i.e. $\mathbb{P}^1(\cdot)=\mathbb{P}(\cdot | A)$. When you take expectation with respect to that, you indicate it by $\mathbb{E}[\cdot|A]$.

In your calculation your indicator function is meaningless. What is the set operation $A | B$?