## Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn't ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at: First year mathematics notes and resources (particularly for the University [...]

## The (Central) Cauchy distribution

The core of this post comes from Mathematical Statistics and Data Analysis by John A. Rice which is a useful resource for subjects such as UCT's STA2004F. Introduction The Cauchy distribution has a number of interesting properties and is considered a pathological (badly behaved) distribution. What is interesting about it is that [...]

## K-means: Intuitions, Maths and Percy Tau

Much of this content is based on lecture slides from slides from Professor David Barber at University College London: resources relating to this can be found at: www.cs.ucl.ac.uk/staff/D.Barber/brml The K-means algorithm The K-means algorithm is one of the simplest unsupervised learning* algorithms. The aim of the K-means algorithm is, given a [...]

## What’s the shortest known Normal Number?

Well, the answer is that it has to be infinitely long, but the question is what is the most compact form of a Normal Number possible. I was motivated to look into this from a lovely Numberphile video about all the real numbers. Normal numbers in base [...]

## p-values (part 2) : p-Hacking Why drinking red wine is not the same as exercising

What is p-hacking? You might have heard about a reproducibility problem with scientific studies. Or you might have heard that drinking a glass of red wine every evening is equivalent to an hour's worth of exercise. Part of the reason that you might have heard about these things is p-hacking: [...]

## A quick argument for why we don’t accept the null hypothesis

Introduction When doing hypothesis testing, an often-repeated rule is 'never accept the null hypothesis'. The reason for this is that we aren't making probability statements about true underlying quantities, rather we are making statements about the observed data, given a hypothesis. We reject the null hypothesis if the observed data [...]

## p-values: an introduction (Part 1)

The starting point This is the first of (at least) 3 posts on p-values. p-values are everywhere in statistics- especially in fields that require experimental design. They are also pretty tricky to get your head around at first. This is because of the nature of classical (frequentist) statistics. So to [...]

## R-squared values for linear regression

What we are talking about Linear regression is a common and useful statistical tool. You will have almost certainly come across it if your studies have presented you with any sort of statistical problems. The pros of regression are that it is relatively easy to implement and that the relationship [...]

## Cantor–Schröder–Bernstein Theorem

Knowledge this posts assumes: What is a set, set cardinality, a function, an image of a function and an injective (one-to-one) function. David Hilbert imagines a hotel with an infinite number of rooms. In this hotel, each room can only be occupied by one guest, and each room is indeed [...]

## 1.6 Partitions

Recall the relation $latex \equiv \text{ mod} (4) $ on the set $latex \mathbb{ N}.$ One of the equivalence classes is $latex [0] = \{ ..., -8, -4, 0, 4, 8, ...\}$ which is equivalent to writing $latex [0] = [4] = [-4] = [8] = [-8] ...$ We could do this [...]