Math Camp

Math Camp 2023 will be in-person. I will upload my notes in late summer. Math Camp 2022 was hosted virtually via Zoom. As such, all of my lectures were recorded and uploaded. Below, you can find the videos to watch.

I also created a problem set which you can find here: Math Camp Problem Set.

1. Probability Pre-requisites

This video covers the basics of probability theory: outcomes, events, sets, and sigma algebras. I prove the partitioning theorem.

2. Properties of the Probability Function

I define the probability measure, the probability function, and the probability space. I go through the axioms of probability. Lastly, I prove properties of the probability measure. The proofs can also be found here: Proofs.

3. Calculating Probabilities

I cover the principle of equally-likely outcomes, conditional probability, and independence.

4. More Probability

I continue with calculating probabilities by covering and deriving the law of total probability. I also derive Bayes' rule. "Your Turn" question 1 is at the end of this video.

5. Counting

I cover the counting rule, permutations, and combinations. I introduce the Binomial Theorem. I briefly talk about the difference between "with replacement" and "without replacement."

6. Poker Hands

I calculate the probability of receiving a variety of five-card-draw poker hands on the first deal. "Your Turn" question 2 is at the end of this video.

7. Discrete Random Variables

I define a random variable, the cumulative distribution function, and types of random variables. I then define discreteness, the probability mass function, and a number of discrete distributions.

8. Continuous Random Variables

I define continuity, provide an epsilon-delta continutity proof, define the probability density function, define the normalizing constant and kernel, and cover some exmaples of continuous distributions. I also cover transformations.

9. Moments

The last video covers the expected value, variance, and standard deviation of a random variable. I look at the moments of linear transformations. I prove the variance decomposition formula. Lastly, I cover skewness and kurtosis. "Your Turn" question 3 is the proof of the moments of linear transformations theorem. You can find the "Your Turn" solutions here: Solutions