The Optional Stopping/Sampling Theorem

The OST says that given a finite amount of money and a finite time horizon, any betting sequence/strategy cannot improve the expected earnings from a sequence of fair gambles.  More to the point, the theorem is proof that you can’t expect to make money playing a fair game (a martingale) and you can’t bet in such a way that you can expect to beat an unfair game (a supermartingale).  Attached, I’ve written a rundown of the result along with a nice short proof.

The Optional Stopping/Sampling Theorem

Stochastic Process, Optional Stopping Theorem, Expected Value, Conditional ExpectationMartingale

Riemann Zeta (2)

Linked is a proof written on the value of the Riemann Zeta function at 2 (aka the Basel Problem). My solution utilizes the Fourier transform and the strong form of L’Hopital’s rule, as well as hyperbolic trig identities, etc… While it is indeed a correct derivation, there are simpler solutions to the problem. This solution is interesting though. …i think.

Riemann Zeta (2)

Basel Problem, Poisson Summation Formula, Fourier Transform, L’Hopital’s rule, Taylor Series

Math Heroes!

A collection of homemade images featuring history’s great mathematicians – Beautiful!

Proof of the Least Upper Bound Property… but not really.

The least upper bound property is a property of the real numbers which distinguishes it from other ordered fields. The property states that every bounded subset of R has a least upper bound in R. The LUBP is an assumed property of the real numbers as it can not be proven from the ordered field axioms and is important because it is essential in the proof that a sequence is Cauchy iff it is convergent (which is to say that the real numbers are a complete ordered field). However, the LUBP can be proven if we assume that a sequence is Cauchy iff it is convergent. Linked is an interesting and fun proof of such, written as an undergraduate.

Proof of the Least Upper Bound Property… but not really