The notation and concepts in this exercise are from Appendix...

The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so $\Omega=\left\{\omega_1, \omega_2, \omega_3\right\}$ with $\mathbb{P}\left(\left\{\omega_1\right\}\right)=\mathbb{P}\left(\left\{\omega_2\right\}\right)=\mathbb{P}\left(\left\{\omega_3\right\}\right)=1 / 3$. Let $\mathcal{G}$ be the collection of all subsets of $\Omega$ : $$ \mathcal{G}=\left\{\emptyset,\left\{\omega_1\right\},\left\{\omega_2\right\},\left\{\omega_3\right\},\left\{\omega_1, \omega_2\right\},\left\{\omega_1, \omega_3\right\},\left\{\omega_2, \omega_3\right\}, \Omega\right\} $$ Let $\bar{x}$ and $\tilde{y}$ be random variables, and set $a_i=\bar{x}\left(\omega_i\right)$ for $i=1,2,3$. Suppose $\bar{y}\left(\omega_1\right)=b_1$ and $\bar{y}\left(\omega_2\right)=\bar{y}\left(\omega_3\right)=b_2 \neq b_1$. (a) What is $\operatorname{prob}\left(\tilde{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ? (b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ? (c) What is the $\sigma$-field generated by $\bar{y}$ ? (a) What is $\operatorname{prob}\left(\bar{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ? (b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ? (c) What is the $\sigma$-field generated by $\bar{y}$ ?

The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so $\Omega=\left\{\omega_1, \omega_2, \omega_3\right\}$ with $\mathbb{P}\left(\left\{\omega_1\right\}\right)=\mathbb{P}\left(\left\{\omega_2\right\}\right)=\mathbb{P}\left(\left\{\omega_3\right\}\right)=1 / 3$. Let $\mathcal{G}$ be the collection of all subsets of $\Omega$ :
$$
\mathcal{G}=\left\{\emptyset,\left\{\omega_1\right\},\left\{\omega_2\right\},\left\{\omega_3\right\},\left\{\omega_1, \omega_2\right\},\left\{\omega_1, \omega_3\right\},\left\{\omega_2, \omega_3\right\}, \Omega\right\}
$$
Let $\bar{x}$ and $\tilde{y}$ be random variables, and set $a_i=\bar{x}\left(\omega_i\right)$ for $i=1,2,3$. Suppose $\bar{y}\left(\omega_1\right)=b_1$ and $\bar{y}\left(\omega_2\right)=\bar{y}\left(\omega_3\right)=b_2 \neq b_1$.
(a) What is $\operatorname{prob}\left(\tilde{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ?
(b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ?
(c) What is the $\sigma$-field generated by $\bar{y}$ ?
(a) What is $\operatorname{prob}\left(\bar{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ?
(b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ?
(c) What is the $\sigma$-field generated by $\bar{y}$ ?

Key Concepts

Sample Space

The sample space is the set of all possible outcomes of a random experiment. It provides the foundation for defining events and assigning probabilities, ensuring that every outcome is considered when analyzing the behavior of random phenomena.

Sigma-Field

A sigma-field (or sigma-algebra) is a collection of subsets of the sample space that includes the empty set, is closed under complementation, and is closed under countable unions. This structure is essential in measure theory as it helps define measurable events to which probabilities can be consistently assigned.

Probability Measure

A probability measure is a function that assigns a non-negative real number to events in a sigma-field and satisfies axioms such as normalization (the probability of the entire space is 1) and countable additivity. It formalizes the concept of likelihood in a probability space.

Random Variables

A random variable is a measurable function from the sample space to the real numbers (or another measurable space), allowing outcomes to be quantified and facilitating the study of their distributions and expected values within the framework of probability.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has occurred. It refines the probability measure by restricting the sample space to the known event and is fundamental in updating beliefs based on new information.

Conditional Expectation

Conditional expectation is the expected value of a random variable given the occurrence of a specific event or the value of another random variable. It serves as a best approximation of the random variable under the information provided by the conditioning event or sigma-field.

Sigma-Field Generated by a Random Variable

The sigma-field generated by a random variable is the collection of all events that can be described in terms of that random variable. It represents the information contained in the random variable and is crucial in defining concepts like conditional expectation, as it determines which events are observable or measurable relative to that variable.

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