Consider the following probability space \((\Omega,P,\mathcal{F})\) where \(\Omega=\big\{ (\omega_0,\omega_1) : \omega_i \in \{-1,0,1\} \big\}\). Take of each of the \(\omega_i\) to be mutually independent with \(\mathbf P(\omega_i=0)=\frac12\) and \(\mathbf P(\omega_i=\pm 1)=\frac14\). (\(\mathcal{F}\) is just the \(\sigma\)-algebra generated by the collection of single points, but this is not important)

For \(n=0\) or \(1\) define the random variables \(X_n\) by  \(X_0=\omega_0\) and \(X_1=\omega_0 \omega_1\).

1. What is \(\mathbf P(X_1=1)\) ? What is \(\mathbf E X_0\) ?
2. Let \(A\) be the event that \(\{X_1\neq 0 \}\). What is \(\sigma(A)\) ?
3. What is \(\sigma(X_0)\)( and \(\sigma(X_1)\) ?
4. What is \(\mathbf E(X_1 | A)\) ? What is \(E(X_1 | X_0)\) ?