Lecture 25: Order Statistics and Conditional Expectation | Statistics 110 | Summary and Q&A

TL;DR

The beta and gamma distributions are connected through a famous example called the "bank and post office" which helps understand the distribution of waiting times.

Transcript

So we were talking about the beta distribution and the gamma distribution separately. And I mentioned that they're connected but we haven't seen, how is beta connected to gamma other than being consecutive letters of the Greek alphabet? So there's one very, very famous example that connects the beta and the gamma in a very neat way. So that seemed ... Read More

Key Insights

• 🏣 The beta and gamma distributions are connected through the "bank and post office" example.
• 👨‍💼 The total waiting time is distributed as gamma(a + b, lambda) for the bank and post office example.
• 🏦 The fraction of time spent waiting at the bank is distributed as beta(a, b).
• 👨‍💼 The joint distribution of the waiting time at the bank and the post office is gamma(a + b, lambda) * beta(a, b).
• 😃 The normalizing constant for the beta distribution can be obtained from the marginal PDFs, which is gamma(a + b).
• 🏣 The beta and gamma distributions have a special property that makes them connected and independent in the "bank and post office" example.
• ❓ The connection between the beta and gamma distributions can be used to find expected values and other statistical quantities.
• 🪈 The distribution of order statistics can be found by considering the probability of certain events occurring and using the binomial distribution.

Questions & Answers

Q: What is the distribution of the total waiting time?

The distribution of the total waiting time is gamma(a + b, lambda) for the bank and post office example.

Q: Are the waiting times at the bank and post office independent?

Yes, the waiting times at the bank and post office are assumed to be independent in the bank and post office example.

Q: What is the distribution of the fraction of time spent waiting at the bank?

The distribution of the fraction of time spent waiting at the bank is beta(a, b).

Q: How can the normalizing constant for the beta distribution be obtained?

The normalizing constant for the beta distribution can be obtained by calculating the marginal PDFs and finding their normalizing constant, which is gamma(a + b).

Summary

In this video, the presenter starts by discussing the connection between the beta distribution and the gamma distribution. They then introduce the concept of order statistics and explain how to find the distribution of a specific order statistic. The presenter also discusses conditional expectation and its properties. Finally, they present the "Two Envelope Paradox" and leave viewers to think about it.

Questions & Answers

Q: How are the beta and gamma distributions connected?

The beta and gamma distributions are connected through a special example called the "bank and post office" example. In this example, the waiting time at the bank is modeled by a gamma distribution and the waiting time at the post office is also modeled by a gamma distribution. The total waiting time is then the sum of these two waiting times, which follows a gamma distribution.

Q: How can we find the distribution of the total waiting time in the "bank and post office" example?

If the waiting time at the bank follows a gamma distribution with parameters a and lambda, and the waiting time at the post office follows a gamma distribution with parameters b and lambda, then the distribution of the total waiting time is gamma(a + b, lambda).

Q: What is the fraction of time spent waiting at the bank in the "bank and post office" example?

The fraction of time spent waiting at the bank, denoted as W, is a natural quantity to look at in this example. To find its distribution, we need to know the joint distribution of the total waiting time and the fraction of time spent waiting at the bank. It turns out that the total waiting time and the fraction of time spent waiting at the bank are independent. Therefore, the distribution of W is a beta distribution with parameters a and b.

Q: What is the normalizing constant of the beta distribution?

The normalizing constant of the beta distribution is given by gamma(a + b) / (gamma(a) * gamma(b)), where gamma represents the gamma function.

Q: How can we find the distribution of the jth order statistic in a sample of size n?

To find the distribution of the jth order statistic, we can use the formula f(x) = n(n-1 choose j-1) F(x)^(j-1) (1-F(x))^(n-j), where f(x) is the PDF of the jth order statistic and F(x) is its CDF.

Q: Can you provide an example of the distribution of an order statistic?

One example is the distribution of the jth order statistic in a sample of n iid uniform random variables between 0 and 1. In this case, the jth order statistic follows a beta distribution with parameters j and n-j+1.

Q: What is the Two Envelope Paradox?

The Two Envelope Paradox is a situation where you are given two envelopes, each containing a check for some amount of money. One envelope has exactly twice as much money as the other. You pick one envelope and, after opening it, you are given the option to switch to the other envelope. The paradox arises when you realize that on average, the other envelope contains more money, leading to the conclusion that you should switch. However, this reasoning leads to an infinite loop of switching back and forth, which is counterintuitive.

Summary & Key Takeaways

• The beta and gamma distributions are connected through a real-life example called the "bank and post office" where waiting times are modeled by gamma distributions.

• The total waiting time is distributed as gamma(a + b, lambda).

• The fraction of time spent waiting at the bank is distributed as beta(a, b).

• The joint distribution of the waiting time at the bank and the post office is gamma(a + b, lambda) * beta(a, b).