Lecture 26: Conditional Expectation Continued | Statistics 110 | Summary and Q&A

TL;DR

Conditional expectation allows us to predict the value of a random variable given another random variable, and can be used to solve problems like the Two-Envelope Problem.

Key Insights

• 👻 Conditional expectation allows us to predict the value of a random variable given the value of another random variable.
• 🤑 The Two-Envelope Problem involves analyzing different arguments and using conditional expectation to determine the expected value of the money in the envelopes.
• 💌 The symmetry argument in the Two-Envelope Problem suggests that there is no inherent difference between the envelopes and supports the idea that the expected values should be equal.

Transcript

Last time I left you with a cliffhanger, right? So we better try to resolve that, the two envelope problem. So, I'll remind you what the problem is, it's very simple to state, but not so easy to resolve. So the problem is just, we have two envelopes, envelope one, envelope two. They look identical, and suppose they are X dollars in here and Y dolla... Read More

Questions & Answers

Q: What is the Two-Envelope Problem?

The Two-Envelope Problem involves two envelopes, one with twice as much money as the other, and the goal is to determine which envelope has more money. It is a paradoxical problem with different arguments and no clear solution.

Q: How does conditional expectation help in the Two-Envelope Problem?

Conditional expectation is used to analyze the different arguments in the Two-Envelope Problem and determine the expected value of the money in the envelopes based on conditional information. It helps to resolve the contradiction in the arguments.

Q: Why is the symmetry argument important in the analysis of the Two-Envelope Problem?

The symmetry argument states that there is no inherent difference between the envelopes, and therefore, it is hard to argue against it. This argument supports the idea that the expected value of the money in the envelopes should be equal.

Q: What is the mistake in the argument involving conditional expectation in the Two-Envelope Problem?

The mistake is in plugging in the conditional information without considering the condition itself. This assumes that the conditioning information can be forgotten, leading to a contradiction. The correct interpretation is that the information should still be present and affect the conditional expectation.

Summary

This video discusses the two envelope problem and conditional expectations. The two envelope problem involves two envelopes, each containing a random amount of money. One envelope has twice as much money as the other, but you do not know which one. The video explores two arguments about which envelope is better, but concludes that there is no clear answer. The video also introduces conditional expectations, explaining how to compute them and their properties.

Questions & Answers

Q: What is the two envelope problem?

The two envelope problem involves two envelopes, each containing a random amount of money. One envelope has twice as much money as the other, but you do not know which one.

Q: What are the two competing arguments in the two envelope problem?

Argument one is based on symmetry, stating that the expected amount of money in each envelope is equal. Argument two is based on the law of total probability, stating that the expected amount of money in each envelope is five-fourths of the expected amount in the chosen envelope.

Q: Why is argument one considered a strong argument in the two envelope problem?

Argument one is considered strong because it is based on the principle of symmetry, which suggests that there should be no difference in the expected amount of money in each envelope.

Q: How is argument two computed in the two envelope problem?

Argument two uses the law of total probability to compute the expected amount of money in each envelope. By conditioning on the possible values for the amount of money in the other envelope, argument two calculates the expected value to be five-fourths of the expected amount in the chosen envelope.

Q: Can both argument one and argument two be true in the two envelope problem?

No, argument one and argument two cannot both be true because they lead to contradictory conclusions. Since argument one is based on symmetry and argument two is based on the law of total probability, one of them must be false.

Q: What is a common mistake when using conditioning in probability problems?

A common mistake when using conditioning is to forget about the information that was conditioned on. In the example of the two envelope problem, the mistake is to plug in the information about the amount of money in the other envelope but then forget to use that information in the calculation of conditional expectations.

Q: How can the mistake of forgetting conditioned information be applied in the two envelope problem?

In the two envelope problem, the mistake of forgetting conditioned information occurs when calculating the expected amount of money in each envelope. The correct calculation requires using the conditional expectations of the amount of money in the other envelope, but the mistake occurs when this conditional information is forgotten and the calculations are simplified incorrectly.

Q: What is the relationship between conditional expectations and indicator random variables in the two envelope problem?

In the two envelope problem, conditional expectations can be related to indicator random variables. The indicator random variable represents whether one envelope has more money than the other. The relationship between the conditional expectations of the amount of money in each envelope and the indicator random variable shows that the two are dependent, which is surprising at first but makes sense when considering the clumping of certain patterns in the sequence of coin flips.

Q: What is the coin flipping problem discussed in the video?

The coin flipping problem involves flipping a fair coin repeatedly until a certain pattern is observed. The video explores two specific patterns: heads followed by tails (HT) and heads twice in a row (HH). The goal is to find the expected number of coin flips until each pattern occurs.

Q: How are conditional expectations used to solve the coin flipping problem?

Conditional expectations are used to solve the coin flipping problem by breaking down the problem into simpler pieces. By conditioning on the first flip and observing the resulting conditional expectations, the expected number of flips until each pattern occurs can be calculated.

Q: What is the expected number of flips until the HT pattern occurs in the coin flipping problem?

The expected number of flips until the HT pattern occurs is 4.

Takeaways

The two envelope problem demonstrates the importance of properly using conditional expectations and considering all relevant information in probability problems. Conditional expectations can help simplify complex problems by breaking them down into smaller, more manageable pieces. It is also crucial to pay attention to assumptions and dependencies in probability problems to ensure accurate calculations and avoid common mistakes. The coin flipping problem illustrates the applicability of conditional expectations in real-world scenarios, such as genetics, where patterns in sequences are analyzed. Overall, understanding conditional expectations and their properties is essential for solving a variety of probability problems.

Summary & Key Takeaways

• Conditional expectation is a way to predict the value of a random variable given the value of another random variable. It is calculated based on the conditional probability of the two variables.

• The Two-Envelope Problem involves two envelopes with an unknown amount of money, where one envelope has twice as much as the other. Conditional expectation is used to analyze different arguments and resolve the problem.

• In the Two-Envelope Problem, when conditioning on one envelope, the other envelope can have either twice as much or half as much money. By averaging the expectations, it suggests that one envelope is better, but the same argument can be applied to the other envelope.

• The key insight is that in the Two-Envelope Problem, the mistaken step is when the conditional information is plugged in without considering the condition itself. This leads to a contradiction and shows that the two arguments cannot both be true.