This post is about some fundamental concepts in classical (or frequentist) statistics: inference and hypothesis testing. A while back, I came to the realization that I didn't have a good intuition of these concepts (at least not to my liking) beyond the mechanical nature of applying them. What was missing was how they related to a probabilistic view of the subject. This bothered me since having a good intuition about a subject is probably the most useful (and fun!) part of learning a subject. So this post is a result of my re-education on these topics. Enjoy!
In this post, I'm going to continue on the same theme from the last post: random sampling. We're going to look at two methods for sampling a distribution: rejection sampling and Markov Chain Monte Carlo Methods (MCMC) using the Metropolis Hastings algorithm. As usual, I'll be providing a mix of intuitive explanations, theory and some examples with code. Hopefully, this will help explain a relatively straight-forward topic that is frequently presented in a complex way.
One of the most common probability distributions is the normal (or Gaussian) distribution. Many natural phenomena can be modeled using a normal distribution. It's also of great importance due to its relation to the Central Limit Theorem.
In this post, we'll be reviewing the normal distribution and looking at how to draw samples from it using two methods. The first method using the central limit theorem, and the second method using the Box-Muller transform. As usual, some brief coverage of the mathematics and code will be included to help drive intuition.
My last post was about some common mistakes when betting or gambling, even with a basic understanding of probability. This post is going to talk about the other side: optimal betting strategies using some very interesting results from some very famous mathematicians in the 50s and 60s. I'll spend a bit of time introducing some new concepts (at least to me), setting up the problem and digging into some of the math. We'll be looking at it from the lens of our simplest probability problem: the coin flip. A note: I will not be covering the part that shows you how to make a fortune -- that's an exercise best left to the reader.
Games and gambling have been part of human cultures around the world for millennia. Nowadays, the connection between games of chance and mathematics (in particular probability) are so obvious that it is taught to school children. However, the mathematics of games and gambling only started to formally develop in the 17th century with the works of multiple mathematicians such as Fermat and Pascal. It is then no wonder that many incorrect beliefs around gambling have formed that are "intuitive" from a layman's perspective but fail to pass muster when applying the rigor of mathematics.
In this post, I want to discuss how surprisingly easy it is to be fooled into the wrong line of thinking even when approaching it using mathematics. We'll take a look from both a theoretical (mathematics) point of view looking at topics such as the Gambler's Fallacy and the law of small numbers as well as do some simulations using code to gain some insight into the problem. This post was inspired by a paper I recently came across a paper by Miller and Sanjurjo that explains the surprising result of how easily we can be fooled.