Consider a probability space $\Omega$ with four elements, which we call $a, b, c$, and $d$ (i.e., $\Omega=\{a, b, c, d\}$ ). The $\sigma$-algebra $\mathcal{F}$ is the collection of all subsets of $\Omega$; i.e., the sets in $\mathcal{F}$ are
$$
\begin{aligned}
& \Omega,\{a, b, c\},\{a, b, d\},\{a, c, d\},\{b, c, d\}, \\
& \{a, b\},\{a, c\},\{a, d\},\{b, c\},\{b, d\},\{c, d\}, \\
& \{a\},\{b\},\{c\},\{d\}, \emptyset .
\end{aligned}
$$
We define a probability measure $\mathbb{P}$ by specifying that
$$
\mathbb{P}\{a\}=\frac{1}{6}, \mathbb{P}\{b\}=\frac{1}{3}, \mathbb{P}\{c\}=\frac{1}{4}, \mathbb{P}\{d\}=\frac{1}{4},
$$
and, as usual, the probability of every other set in $\mathcal{F}$ is the sum of the probabilities of the elements in the set, e.g., $\mathbb{P}\{a, b, c\}=\mathbb{P}\{a\}+\mathbb{P}\{b\}+\mathbb{P}\{c\}=\frac{3}{4}$.
We next define two random variables, $X$ and $Y$, by the formulas
$$
\begin{aligned}
& X(a)=1, X(b)=1, X(c)=-1, X(d)=-1, \\
& Y(a)=1, Y(b)=-1, Y(c)=1, Y(d)=-1 .
\end{aligned}
$$
We then define $Z=X+Y$.
(i) List the sets in $\sigma(X)$.
(ii) Determine $\mathbb{E}[Y \mid X]$ (i.e., specify the values of this random variable for $a$, $b, c$, and $d$ ). Verify that the partial-averaging property is satisfied.
(iii) Determine $\mathbb{E}[Z \mid X]$. Again, verify the partial-averaging property.
(iv) Compute $\mathbb{E}[Z \mid X]-\mathbb{E}[Y \mid X]$. Citing the appropriate properties of conditional expectation from Theorem 2.3.2, explain why you get $X$.