Lecture 15: Midterm Review | Statistics 110 | Summary and Q&A

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April 29, 2013
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Harvard University
Lecture 15: Midterm Review | Statistics 110

TL;DR

The content explores the geometric distribution, expectations, and the universality of the uniform distribution.

Key Insights

• π« The geometric distribution is a useful tool for calculating the expected time to collect a full set of items with different probabilities.
• π₯ The universality of the uniform distribution means that plugging a random variable into its own CDF results in a uniform distribution.
• π Linearity can be used to calculate the expected value of a random variable by expressing it in terms of other random variables.

Transcript

I had a couple requests on reviewing certain things, so I'm gonna do a few. I just picked a few examples that I'll talk about, but if in the meantime you think of anything else you want to ask, then ask at any point, okay? So first, here's kind of a famous example that I wanted to go through anyway, which is good review of geometric distribution an... Read More

Q: How does the coupon collector problem relate to the geometric distribution and expectations?

The coupon collector problem involves collecting a full set of toys with different probabilities. The expected time to collect a full set can be calculated using the geometric distribution, where each trial represents collecting a new toy.

Q: What is the relationship between the uniform distribution and random variables?

When a random variable is plugged into its own cumulative distribution function (CDF), the resulting distribution is uniform. This demonstrates the universality of the uniform distribution.

Q: How can the time to collect a full set of toys be calculated using linearity and the geometric distribution?

By breaking down the time into the components of collecting each new toy, each component can be treated as a geometric random variable. The expected time to collect a full set is then calculated by summing the expected times for each component.

Q: How can the expected value of a random variable be calculated using linearity and symmetry?

By expressing the random variable as the sum or difference of other random variables, the expected value can be calculated using linearity. Additionally, symmetry allows for swapping success and failure in certain distributions.

Summary

In this video, the speaker covers a few different examples related to geometric distribution, a uniform distribution, and the coupon collector problem. They also discuss the universality of the uniform distribution and provide a visual explanation. They then dive into the topic of linearity in probability and discuss the expectations of multiple random variables. They provide examples with geometric distributions and show how to use linearity to calculate the expected time to collect a full set. The video also touches on symmetry, LOTUS (Law of the Unconscious Statistician), and the Poisson distribution.

Q: What is the coupon collector problem?

The coupon collector problem involves collecting a full set of a certain number of different types of toys. Each time a toy is collected, it is a random toy and may be one that has already been collected. The question is how long it will take, on average, to collect the full set of toys.

Q: How can the problem of collecting a full set of toys be approached mathematically?

The problem can be broken down into smaller subproblems by considering the time it takes to collect each new toy type. Each subproblem can be modeled as a geometric distribution, where the probability of success is the chance of collecting a new toy type and the probability of failure is the chance of collecting a toy type that has already been collected. By summing up the expectations of these subproblems, we can find the expected time to collect a full set.

Q: How can linearity be used in solving problems with multiple random variables?

Linearity allows us to calculate the expectation of the sum of multiple random variables by taking the sum of their individual expectations. This holds even if the random variables are dependent. In the case of the coupon collector problem, we can break down the problem into subproblems and calculate the expectation of each subproblem as a geometric distribution. Then, by using linearity, we can find the expected time to collect a full set by summing up the individual expectations.

Q: What is the n harmonic sum in the context of the coupon collector problem?

The n harmonic sum is the sum of the reciprocals of positive integers up to n. It arises in the calculation of the expected time to collect a full set of toys in the coupon collector problem. The n harmonic sum is approximately equal to n times the natural logarithm of n for large values of n.

Q: What is the universality of the uniform distribution?

The universality of the uniform distribution states that if a random variable X follows a certain distribution, then plugging X into its own cumulative distribution function (CDF) will result in a uniform distribution. This means that the probability of being less than or equal to any given value in the uniform distribution is proportional to the length of the interval.

Q: How can the universality of the uniform distribution be explained visually?

The universality of the uniform distribution can be visualized by plotting a cumulative distribution function (CDF) and random points on the x-axis. If a point is chosen randomly on the x-axis according to the original distribution, then its corresponding y-value, obtained by plugging it into the original CDF, will follow a uniform distribution between 0 and 1.

Q: How can the simulation of random variables from a logistics distribution be done?

One way to simulate random variables from a logistics distribution is by using the inverse transform method. First, simulate a random variable U from a uniform distribution. Then, find the inverse CDF of the logistics distribution and plug U into it. The resulting value will be a random variable from the logistics distribution.

Q: What is the symmetry property in probability and how can it be used?

The symmetry property in probability states that certain properties of a distribution remain the same if the roles of success and failure are interchanged. In the context of calculating expectations, this symmetry can be used to simplify calculations. For example, if X, Y, and Z are independent and identically distributed (IID) positive random variables, finding the expected value of (X/(X+Y+Z)) can be simplified by realizing that the expected values of (Y/(X+Y+Z)) and (Z/(X+Y+Z)) will be the same due to symmetry. This allows us to use linearity and symmetry to find the expected value without needing more complex calculations.

Q: What is LOTUS and how can it be used in probability?

LOTUS stands for the Law of the Unconscious Statistician, which allows us to find the expected value of a function of a random variable without explicitly calculating the distribution of the function. LOTUS states that the expected value of a function of a random variable X can be calculated by integrating the function of X multiplied by the probability density function (PDF) or the probability mass function (PMF) of X, depending on whether X is a continuous or discrete random variable, respectively.

Q: How can LOTUS be applied to find the expected value of a function of a random variable?

To apply LOTUS, we need to find the PDF or PMF of the random variable and the function of the random variable that we are interested in. Then, we integrate the product of the function and the PDF or PMF over the domain of the random variable to find the expected value. It is important to correctly match the distribution (PDF or PMF) with the type of random variable (continuous or discrete) and apply the appropriate integration.

Q: How can the PDF of T, the time of the first email, be found?

To find the PDF of T, we first need to find the complementary cumulative distribution function (CCDF), which is the probability that T is greater than t. In this case, T being greater than t means that there are no emails received in the time interval from 0 to t. Since the number of emails received in that interval follows a Poisson distribution with a mean of lambda t, the CCDF is given by e^(-lambda t). Taking the derivative of the CCDF with respect to t then gives the PDF, which is lambda * e^(-lambda t). This is known as the exponential distribution.

Summary & Key Takeaways

• The content starts by discussing the geometric distribution and expectations, using the example of the coupon collector problem.

• The universality of the uniform distribution is then introduced, emphasizing its relationship to random variables.

• The content concludes with an example of swapping success and failure in a binomial distribution.