Problem 24 - A Computational Introduction to Number Theory and Algebra

Question

Let \(B\) be an event with \(\mathrm{P}[B] \neq 0,\) and let \(\left\\{C_{i}\right\\}_{i \in I}\) be a finite, pairwise disjoint family of events whose union contains \(\mathcal{B}\). Again, generalizing the law of total expectation, show that for every real-valued random variable \(X,\) if \(I^{*}:=\left\\{i \in I: \mathrm{P}\left[B \cap C_{i}\right] \neq 0\right\\},\) then we have $$ \mathrm{E}[X \mid \mathcal{B}]=\sum_{i \in I^{*}} \mathrm{E}\left[X \mid \mathcal{B} \cap C_{i}\right] \mathrm{P}\left[\mathcal{C}_{i} \mid \mathcal{B}\right] $$

Step-by-Step Solution

Answer

Question: Show that for every real-valued random variable \(X\), the expectation of \(X\) given \(B\) is equal to the sum of the expectation of \(X\) given \(B \cap C_i\) multiplied by the probability of \(C_i\) given \(B\), for all \(i \in I^*\), where \(\left\\{C_{i}\right\\}_{i \in I}\) is a finite, pairwise disjoint family of events whose union contains event \(B\) and \(I^*\) is a set of indices for which \(\mathrm{P}\left[B \cap C_{i}\right] \neq 0\). Answer: We have shown in the step-by-step solution that the expectation of \(X\) given \(B\) can be expressed as $$ \mathrm{E}[X \mid B] = \sum_{i \in I^*} \mathrm{E}\left[X \mid B \cap C_i\right] \mathrm{P}\left[C_i \mid B\right]. $$ This result generalizes the law of total expectation for real-valued random variables to the case where the event \(B\) is contained in the union of a finite, pairwise disjoint family of events, given that the probabilities \(\mathrm{P}\left[B \cap C_{i}\right] \neq 0\) for all \(i \in I^*\).

Step 1: Write down the definition of conditional expectation

Recall the definition of conditional expectation for a real-valued random variable \(X\) given an event \(B\) with \(\mathrm{P}[B] \neq 0\): $$ \mathrm{E}[X \mid B] = \frac{\mathrm{E}[X \mathbf{1}_B]}{\mathrm{P}[B]} $$ Similarly, we can also write the definition of conditional expectation for \(X\) given the intersection of events \(B\) and \(C_i\): $$ \mathrm{E}[X \mid B \cap C_i] = \frac{\mathrm{E}[X \mathbf{1}_{B \cap C_i}]}{\mathrm{P}[B \cap C_i]} $$

Step 2: Express each term in the given formula

Now let's focus on expressing each term in the given formula. We'll start with the expectation of \(X\) given \(B \cap C_i\) and the probability of \(C_i\) given \(B\): $$ \text{First term: }\mathrm{E}\left[X \mid B \cap C_i\right] = \frac{\mathrm{E}[X \mathbf{1}_{B \cap C_i}]}{\mathrm{P}[B \cap C_i]} $$ $$ \text{Second term: }\mathrm{P}\left[C_i \mid B\right] = \frac{\mathrm{P}[B \cap C_i]}{\mathrm{P}[B]} $$

Step 3: Multiply the terms and sum over all \(i \in I^*\)

Next, let's multiply the two terms for each \(i \in I^*\) and sum them up: $$ \sum_{i \in I^*} \mathrm{E}\left[X \mid B \cap C_i\right] \mathrm{P}\left[C_i \mid B\right] = \sum_{i \in I^*} \frac{\mathrm{E}[X \mathbf{1}_{B \cap C_i}]}{\mathrm{P}[B \cap C_i]} \cdot \frac{\mathrm{P}[B \cap C_i]}{\mathrm{P}[B]} $$ As we can see, the terms \(\mathrm{P}[B \cap C_i]\) cancel out, leaving us with: $$ \sum_{i \in I^*} \frac{\mathrm{E}[X \mathbf{1}_{B \cap C_i}]}{\mathrm{P}[B]} $$

Step 4: Use the properties of \(\mathrm{P}[B]\) and simplify the sum

Since \(\left\\{C_{i}\right\\}_{i \in I}\) are pairwise disjoint and their union contains \(B\), we can write \(\mathrm{P}[B]\) as the sum of probabilities \(\mathrm{P}[B \cap C_i]\) for \(i \in I^*\). Therefore, we can rewrite the expression above as: $$ \sum_{i \in I^*} \mathrm{E}[X \mathbf{1}_{B \cap C_i}] \frac{1}{\sum_{j \in I^*} \mathrm{P}[B \cap C_j]} $$ Now we can change the order of summation and obtain: $$ \frac{\sum_{i \in I^*} \mathrm{E}[X \mathbf{1}_{B \cap C_i}]}{\sum_{j \in I^*} \mathrm{P}[B \cap C_j]} = \frac{\mathrm{E}[X \mathbf{1}_B]}{\mathrm{P}[B]} $$

Step 5: Conclude that the expressions are equal

Finally, we can conclude that the sum of the terms in the given formula is indeed equal to the expectation of \(X\) given \(B\). We have proved that: $$ \mathrm{E}[X \mid B] = \sum_{i \in I^*} \mathrm{E}\left[X \mid B \cap C_i\right] \mathrm{P}\left[C_i \mid B\right] $$

Key Concepts

Law of Total Expectation

One instrumental concept in understanding random variables in probability theory is the Law of Total Expectation. This law tells us that if we want to find the expected value of a random variable, we can do so by breaking it down into conditional expectations across different, non-overlapping scenarios.

The concept applies wonderfully when dealing with complex systems where a certain amount of uncertainty is involved. By partitioning the sample space into several parts (events) and then calculating the expectation within each one, we can sum all these expectations to get the overall expected value.

For example, imagine having a bag full of differently colored balls and we want to calculate the average weight of the balls. Instead of weighing each ball separately, we can group the balls by color, weigh the average for each color group and then calculate the total average considering the proportion of each color group’s balls in the bag. This method is not only efficient but provides a clear step-by-step process of arriving at the expected weight, which parallels how the Law of Total Expectation works in mathematics.

Random Variables

At the heart of probability theory lie random variables, which essentially are functions that assign numerical values to the outcomes of random processes. Think of them as tags or labels that a mathematician sticks onto outcomes, helping to quantify uncertainty.

For instance, when flipping a coin, while the actual event is the coin landing on heads or tails, a random variable could be used to assign a '1' for heads and a '0' for tails, transforming a physical event into a numerical value for further analysis. Random variables can be discrete, taking on a finite set of possible values, or continuous, with values over an entire interval. Understanding random variables is crucial for grasping more complex concepts in probability, such as conditional expectation and the Law of Total Expectation.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and the likelihood of occurrences. This theoretical framework is the foundation for statistical inference and helps us make sense and predictions about chaotic systems.

Through its axioms and various principles, probability theory provides a structured way to deal with uncertainty and calculate the chances of different outcomes. It includes studying the properties of random variables, events, and operations between events, like union and intersection. Insights from probability theory are not only limited to pure math but have profound applications in fields such as finance, engineering, science, and social sciences.

Pairwise Disjoint Events

The concept of pairwise disjoint events, also known as mutually exclusive events, is extremely important when discussing probability theory. Events are considered pairwise disjoint if they cannot occur simultaneously - the occurrence of one event means all other events cannot occur.

For example, in a deck of cards, drawing an ace of spaces and drawing an ace of hearts are disjoint events because you cannot draw both with a single card pick. Knowing whether a set of events are pairwise disjoint or not helps greatly in probability calculations, as it simplifies the structure of the problem, often allowing us to add probabilities directly if we're interested in the likelihood of any one of the events happening.