#proofs

Claims: If \(n\) is coprime with \(a\) and \(b\), then \(n\) is coprime with \(ab\). Firstly let us define what we mean by coprime. If \(n\) and \(a\) are coprime, then $$ gcd(n, a) = 1 $$ Using Euclidean algorithm, we know that if (and only if) the above is true then there must exist integers \(x\) and \(y\) such that $$ ax + ny = 1 $$ So then to prove the result above, let \(v\), \(w\), \(x\), \(y\) be some integer such that [Read More]

In the area of machine learning, and statistical modelling, logistic regression, and the use of grouping objects in groups is extremely important. There are plenty of documented ways to access model suitability, primarily dealing with false positive and true positive ratios. But when items are difficult (or expensive) to access and the prevailence of type III errors (correct classification for the wrong reasons, which strangely enough is quite important in some areas), different methods have to be employed to think about model fitness. [Read More]

Kantorovich Inequality is used to show linear convergence for steepest descent algorithm (for the quadratic case). This result is important in some optimization algorithms. Kantorovich Inequality Suppose \(x_1 < x_2 < … < x_n \) are given positive numbers. Let \(\lambda_1, … , \lambda_n \geq 0\) and \(\sum_{i = 1}^n \lambda_i = 1\) then $$ \left( \sum_{i = 1}^n \lambda_i x_i \right) \left( \sum_{i = 1}^n \lambda_i x_i^{-1} \right) \leq \frac{1}{4} \frac{(x_1 + x_n)^2}{x_1 x_n}$$ [Read More]

If \(X~LN(\mu,\sigma^2)\) then \(ln(X)=Y\) is \(Y\) is distributed \(N(\mu,\sigma^2)\).

Derivation of log-normal distribution

\(\begin{align}
Pr(X < k) &= Pr(e^{Y} < k) \\
&= Pr(Y < ln(k)) \\
&= \int_{\infty}^{ln(k)} \frac{1}{\sqrt{2\pi \sigma}} e^{- \frac{(Y-\mu)^2}{2\sigma^2}} dy \\
&= \int_{\infty}^{ln(k)} \frac{1}{\sqrt{2\pi \sigma}} e^{- \frac{(ln(x)-\mu)^2}{2\sigma^2}} \frac{1}{x} \frac{dx}{dy}dy \\
&= \int_{\infty}^{ln(k)} \frac{1}{x\sqrt{2\pi \sigma}} e^{- \frac{(ln(x)-\mu)^2}{2\sigma^2}} dx \end{align}\)

Rigour is lost in the real world. But that does not mean I shouldn’t continue to pursue and practise my mathematical skills. Just like a programmer might learn to code by writing programs, to improve my mathematical thinking I must read other people’s proofs and write my own. (emphasis mine) Adapted from “Why there is no Hitchhiker’s Guide to Mathematics for Programmers”. So perhaps here will be a collection of (mostly other people’s) proofs, until I feel confident writing my own. [Read More]