The past week has consisted of mathematics, more mathematics, and even more mathematics. I had the most insightful conversation a week ago with a fellow quant enthusiast, a bit more accomplished than I am, who told me straight up: look, it’s cool and all to be able to speak the language of the quants, talking about option pricing, Black-Scholes theorems, Greeks, stochastic differential equations, Markov chains, and geometric Brownian motions, but the reality is that some people at his firm can hardly talk about basic logarithmic properties.
This was quite a wake-up call for me. I’ve really been investing myself in the mathematics of quantitative finance, but that is all applied. It comes from mathematics that are supposed to be the base, such as statistics, probability theory, combinatorics, number theory, calculus, and linear algebra. So this week marks the first week that I’ve really invested myself into that foundation. I was on break, but with the help of All of Statistics, I truly began the long road of building a real base in these complex subjects.
I started by getting set with set theory (no pun intended) and spent about four hours just really understanding the notation and what it all means. I then moved on to probability and, with the help of the book and videos from Very Normal, I started to grasp the basics in a rigorous fashion. School often teaches probability like it’s a collection of discrete theorems, but in reality, it is a continuous flow, every theorem is derived from those that came before it.
I then hit the first new concept: probability as a function. In other words, the random variable \(X\), the cumulative distribution function \(CDF\), and the probability density function \(PDF\). This is the beginning of the real foundation of statistics. It looks like Greek at first, but then I remembered that Greeks are Delta, Gamma, Beta, and so forth (lol), which I got to later…
I then went through the basic random variables and distributions: the Binomial Distribution, the Poisson Distribution (I had my niece explain how to pronounce that… still takes me at least three tries every time), the Bernoulli Distribution, and the infamous Normal Distribution.
Next, I jumped into expectation, variance, covariance, and correlation (although I’m still a bit confused about some of the theorems). I skipped multivariable probability for now, as I’m not yet mathematically advanced enough to fully understand it, but I did read through it. I continued and read through the entire book, only truly summarizing up until statistics began. I feel like I now have a much better grasp of most of the core concepts.
Then I jumped into linear algebra and calculus, as these are essential for actually understanding statistics. I learned about all types of matrices, integrals, and differential equations (although I’m still extremely confused about ODEs and PDEs, but I’ll get there). I really enjoyed reading about all of it and trying every type.
I also explored more niche areas of mathematics like combinatorics, permutations, and some recursion. I enjoyed the practical applications of these topics.
I watched an awesome video about inference and hypothesis testing, including the null hypothesis and \(p\)-values.
Today I’m back home, luckily, so now I can apply all of this in code and really set it in. I’ll be attempting some cool machine learning using these techniques, like regression and classification, \(k\)-means, \(k\)-nearest neighbor, and more. Stay tuned and see you next time 🙂
