# Definition of expectation $\mathbb{E}_{\mu}$ ( over an initial distribution $\mu )$

expected valueprobabilityrandom variablesstatistics

Hello Math StackExchange!

I am very frustrated trying to find the definition of $$\mathbb{E}_{\mu}$$ (apparently it means expectation with initial distribution). I have tried to search on many different platforms and books , but am in vain.

Also, is it true that $$\mathbb{E}_{\mu}[X] = \sum_{s} \mu(s)\hspace{1mm} \mathbb{E} [X_{n} | X_0 = s]$$ ?

#### Best Answer

After conversation with @nejimbam, I used the definition from p. 13 of the notes I shared (https://www.math.bgu.ac.il/~yadina/RWnotes.pdf).

$$\mathbb{E}_{\mu}[X] = \sum_{x} \mathbb{P}_{\mu}[X=x] \cdot x \hspace{45mm} \text{(By def. of \mathbb{E}_{\mu})}\\ \hspace{11mm}= \sum_x \big[ \sum_s\mu(s) \hspace{1mm} \mathbb{P}[X=x | X_0 = s] \big] \cdot x \hspace{13mm} \text{(By def. of \mathbb{P}_{\mu} as shared in the notes)}\\ \hspace{11mm}= \sum_x \sum_s\mu(s) \hspace{1mm} \mathbb{P}[X=x | X_0 = s]\cdot x \hspace{17mm} \text{(Distributivity of Summation)}\\ \hspace{11mm}= \sum_s \sum_x \mu(s)\hspace{1mm} \mathbb{P}[X=x | X_0 = s]\cdot x \hspace{17mm} \text{(Switching order of Sums)} \\ \hspace{11mm}= \sum_s \mu(s) \sum_x\mathbb{P}[X=x | X_0 = s]\cdot x \hspace{17mm} \text{(Rearranging)}\\ \hspace{11mm}=\sum_s \mu(s) \hspace{1mm} \mathbb{E}[X=x | X_0 = s] \hspace{29mm} \text{(By def. of \mathbb{E})}$$

I am sorry if this was trivial and “expected” to be known (pun is intended here), but I prefer clear, rigorous definitions / derivations.