Consider the probability space $(\Omega_3,\mathcal{F}_3,\mathbb{P})$ where the outcome space $\Omega_3$ is all sequences of three coin tosses, the $\sigma$-algebra $\mathcal{F}_3$ is all subsets of $\Omega_3$, and the probability measure is generated by a fair coin. Suppose that $Y_n=Y_{n-1}+X_n$ where $X_n=1$ if the $n^{th}$ toss is heads and $X_n=-1$ if the $n^{th}$ toss is tails. Define $$ V_n = \left\{ \begin{array}{lr} Y_n & if \;Y_m\geq0\;for\;0\leq m \leq n\\ 0 &otherwise \end{array} \right.$$ 1. What are the elements of the $\sigma$-algebra $\sigma(Y_2)$?
2.Compute the generalized conditional expectation $\mathbb{E}(V_3\;|\;\sigma(Y_2))$.
I've been messing with this problem but am not entirely sure I know what I'm doing. Can someone give a solution? Here is what I came up with. Define the sets $$A_{HH}=\{HHH,HHT\},\;A_{HT\cup TH}=\{HTH,HTT,THH,THT\},\;A_{TT}=\{TTT,TTH\}$$ It seems to me that $\sigma(Y_2)$ is all complements and unions of $$\{\emptyset,A_{HH},A_{HT\cup TH},A_{TT},\Omega_3\}$$ As far as the generalized conditional expectation I get $$\mathbb{E}(V_3\;|\;A_{HH})=\frac{1}{2}4+\frac{1}{2}2=3,$$ and similarly $\;\mathbb{E}(V_3\;|\;A_{HT \cup TH})=1,\;\mathbb{E}(V_3\;|\;A_{TT})=0$.
Yes. And all this, since you are asked $E(V_3\mid Y_2)$, should be summarized as $$E(V_3\mid Y_2)=\tfrac12Y_2(Y_2+1).$$