## Inequalities

We consider the sum of a random number of random variable (e.g., with customers in a store). We then introduce 4 useful inequalities: Cauchy-Schwarz, Jensen, Markov, and Chebyshev. It’s a statistics lecture at Harvard University.

## Gamma distribution and Poisson process

We introduce the Gamma distribution and discuss the connection between the Gamma distribution and Poisson processes. It’s a statistics lecture at Harvard University.

## Beta distribution

We introduce the Beta distribution and show how it is the conjugate prior for the Binomial, and discuss Bayes’ billiards. Stephen Blyth at Harvard University then gives examples of how probability is used in finance.

## Midterm Review

We work through some extra examples, such as the coupon collector problem, an example of Universality of the Uniform, an example of LOTUS, and a Poisson process example. It’s a statistics lecture at Harvard University.

## The Poisson distribution

We introduce the Poisson distribution, which is arguably the most important discrete distribution in all of statistics. We explore its uses as an approximate distribution and its connections with the Binomial. It’s a statistics lecture at Harvard University.

## Gambler’s Ruin and Random Variables

We analyze the gambler’s ruin problem, in which two gamblers bet with each other until one goes broke. We then introduce random variables, which are essential in statistics and for the rest of the course, and start on the Bernoulli and Binomial distributions. It’s a statistics lecture at Harvard University.

## Conditional Probability

We introduce conditional probability, independence of events, and Bayes’ rule. It’s a statistics lecture at Harvard University.

## Normal distribution

It’s a lecture on statistics at Harvard University. We introduce the Normal distribution, which is the most famous, important, and widely-used distribution in all of statistics.

## Story Proofs, Axioms of Probability

It’s a lecture on statistics at Harvard University. We fill in the “Bose-Einstein” entry of the sampling table, and discuss story proofs. For example, proving Vandermonde’s identity with a story is easier and more insightful than going through a tedious algebraic derivation. We then introduce the axioms of probability.