Graduating with a Bachelor’s in (Applied) Mathematics is something I’m very proud of. I don’t remember being naturally talented in any capacity at mathematics (Abysmal grades in high school mathematics, failed pre-calculus several times, and got a ‘C’ in Calculus I) but somehow I stuck it out and got better at math, and got OK … Continue reading Lessons from a Math Degree →
I. Take an immortal monkey, and have it bang on a typewriter from now until the end of time, then eventually the monkey would produce the complete works of Shakespeare. The saying goes something like that. Human’s aren’t really good at imagining things that are infinite, so while the saying appeals to some sense of … Continue reading How to (Randomly) Write Shakespeare →
My friend tested positive for Tuberculosis. She is young, a vegetarian, and exercises every day. She has good grades and wants to go to medical school. And now she’s going to die. Actually, probably not, TB is pretty treatable, but it certainly doesn’t look like any fun. But her anxiety over the positive test is … Continue reading Queer uses for Probability Theory, or, My Friend Might Be Sick So …
Also known as Rudin’s Infamous Tiny Torture Box. Trigger Warning: Baby Rudin. I. For my undergrad, I majored in Applied Mathematics at a university that was fairly well known for its mathematics department. One of the constants in “Applied Mathematics” is the use of calculus for a variety of applications. “The real world”. In statistics and … Continue reading Book Review: Principles of Mathematic…
Trigger warning: I don’t do a lot of editing, since it’s my least favorite thing, so I apologize for the stream of consciousness flow of this essay. Also, at nearly 5000 words, the essay is long. Also, if I had to say so myself, I think sections I and II are weaker than III and … Continue reading Your hypothesis test sucks inside the Garden of Forking Paths →
Disclaimer: I am not a financial expert. I am a student in statistics and probability. What follows is some exposition on an exercise found in one of my textbooks. It was not a trading book, but rather a book on actuarial models for insurance. I thought it was interesting, and decided to write about it. … Continue reading Should you invest in Google or Bitcoin? (VaR Risk and Utility) →
The statement of the Monotone Convergence Theorem reads: Suppose that the set is measurable, and let be a sequence of non-negative, measurable functions such that, for , Let be defined by as . Then, In this proof, Fatou’s lemma will be assumed. Notice that implies that and so by Fatou’s lemma, for Now, since , … Continue reading Proving Monotone Convergence Theorem with Fatou’s Lemma →
Define the sets and . Let be, of course, the outer Lebesgue measure. Suppose that we have a set . In this case, it is possible to cover with a sequence of elementary sets, that is . Note that since is elementary, they may be written as a finite union of disjoint intervals , , … Continue reading The Outer Lebesgue Measure is Translation Invariant and scales nicely with Dilation. →
It will be shown that in , for any , there is a dense open set such that , where is the outer Lebesgue measure. Consider the set of rationals, , which is known to be countable. So, the elements of the rational numbers may be enumerated so that . Now, fix and around each … Continue reading A dense open set in the reals, with small size →
