Independence of random variables can be affected by changes...

Independence of random variables can be affected by changes of measure. To illustrate this point, consider the space of two coin tosses $\Omega_2=\{H H, H T, T H, T T\}$, and let stock prices be given by $$ \begin{aligned} & S_0=4, S_1(H)=8, S_1(T)=2, \\ & S_2(H H)=16, S_2(H T)=S_2(T H)=4, S_2(T T)=1 . \end{aligned} $$ Consider two probability measures given by $$ \begin{aligned} & \tilde{\mathbb{P}}(H H)=\frac{1}{4}, \tilde{\mathbb{P}}(H T)=\frac{1}{4}, \tilde{\mathbb{P}}(T H)=\frac{1}{4}, \tilde{\mathbb{P}}(T T)=\frac{1}{4}, \\ & \mathbb{P}(H H)=\frac{1}{9}, \mathbb{P}(H T)=\frac{2}{9}, \mathbb{P}(T H)=\frac{2}{9}, \mathbb{P}(T T)=\frac{1}{9} . \end{aligned} $$ Define the random variable $$ X=\left\{\begin{array}{l} 1 \text { if } S_2=4, \\ 0 \text { if } S_2 \neq 4 . \end{array}\right. $$ 78 2 Information and Conditioning (i) List all the sets in $\sigma(X)$. (ii) List all the sets in $\sigma\left(S_1\right)$. (iii) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are independent under the probability measure $\tilde{\mathbb{P}}$. (iv) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are not independent under the probability measure $\mathbb{P}$. (v) Under $\mathbb{P}$, we have $\mathbb{P}\left\{S_1=8\right\}=\frac{2}{3}$ and $\mathbb{P}\left\{S_1=2\right\}=\frac{1}{3}$. Explain intuitively why, if you are told that $X=1$, you would want to revise your estimate of the distribution of $S_1$.

Independence of random variables can be affected by changes of measure. To illustrate this point, consider the space of two coin tosses $\Omega_2=\{H H, H T, T H, T T\}$, and let stock prices be given by
$$
\begin{aligned}
& S_0=4, S_1(H)=8, S_1(T)=2, \\
& S_2(H H)=16, S_2(H T)=S_2(T H)=4, S_2(T T)=1 .
\end{aligned}
$$

Consider two probability measures given by
$$
\begin{aligned}
& \tilde{\mathbb{P}}(H H)=\frac{1}{4}, \tilde{\mathbb{P}}(H T)=\frac{1}{4}, \tilde{\mathbb{P}}(T H)=\frac{1}{4}, \tilde{\mathbb{P}}(T T)=\frac{1}{4}, \\
& \mathbb{P}(H H)=\frac{1}{9}, \mathbb{P}(H T)=\frac{2}{9}, \mathbb{P}(T H)=\frac{2}{9}, \mathbb{P}(T T)=\frac{1}{9} .
\end{aligned}
$$

Define the random variable
$$
X=\left\{\begin{array}{l}
1 \text { if } S_2=4, \\
0 \text { if } S_2 \neq 4 .
\end{array}\right.
$$
78
2 Information and Conditioning
(i) List all the sets in $\sigma(X)$.
(ii) List all the sets in $\sigma\left(S_1\right)$.
(iii) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are independent under the probability measure $\tilde{\mathbb{P}}$.
(iv) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are not independent under the probability measure $\mathbb{P}$.
(v) Under $\mathbb{P}$, we have $\mathbb{P}\left\{S_1=8\right\}=\frac{2}{3}$ and $\mathbb{P}\left\{S_1=2\right\}=\frac{1}{3}$. Explain intuitively why, if you are told that $X=1$, you would want to revise your estimate of the distribution of $S_1$.

Key Concepts

Sigma-Algebra Generated by a Random Variable

A sigma-algebra generated by a random variable is the collection of all events that can be described in terms of that variable. It encapsulates the measurable information available from the variable and is constructed by taking all inverse images of Borel sets under the random variable. This concept is essential in probability theory as it allows us to formalize the notion of knowing the outcome of a random variable and to work with conditional probabilities and independence.

Independence of Sigma-Algebras

Two sigma-algebras are said to be independent if, for every event in one sigma-algebra and every event in the other, the probability of their intersection equals the product of their probabilities. This concept is an extension of the idea of independence of random variables to collections of events, and it is critical in understanding how information or outcomes associated with different random variables may or may not influence each other.

Probability Measure and Change of Measure

A probability measure assigns probabilities to events in a sigma-algebra, adhering to the axioms of probability. Changing the measure means considering a different probability assignment over the same sample space, which can lead to different conclusions about the independence or dependence of events. This is particularly important when comparing models or understanding how weighting particular outcomes differently can alter statistical relationships between random variables.

Conditional Probability and Information Updating

Conditional probability involves updating the likelihood of an event based on the occurrence of another event. It reflects how new information can revise initial probability assessments. When additional information is received, such as the outcome of a specific random variable, the estimate of the distribution of another variable might change. This updating process is foundational in fields like Bayesian inference and is crucial in understanding how observed data can impact beliefs about underlying probability distributions.

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