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Poisson approximations to the negative binomial distribution

This post is an introduction to the negative binomial distribution and a discussion of different ways of approximating the negative binomial distribution. The negative binomial distribution describes the number of times a coin lands on tails before a certain number of heads are recorded. The distribution depends on two parameters p and r.…

by michaelnhowes

Jigsaw puzzles and the isoperimetric inequality

The 2025 World Jigsaw Puzzle Championship recently took place in Spain. In the individual heats, competitors had to choose their puzzle. Each competitor could either complete a 500-piece rectangular puzzle or a 500-piece circular puzzle. For example, in Round F, puzzlers could choose one of the following two puzzles: There are many things to consider…

by michaelnhowes

The Multivariate Hypergeometric Distribution

The hypergeometric distribution describes the number of white balls in a sample of n balls draw without replacement from an urn with m_W white balls and m_B black balls. The probability of having x white balls in the sample is: \displaystyle{f(x; m_W, m_B, n) = \frac{\binom{m_W}{x} \binom{m_B}{n-x}}{\binom{m_W+m_B}{n}}} This is because…

by michaelnhowes

The maximum of geometric random variables

I was working on a problem involving the maximum of a collection of geometric random variables. To state the problem, let (X_i)_{i=1}^n be i.i.d. geometric random variables with success probability p=p(n). Next, define M_n = \max\{X_i : 1 \le i \le n\}. I wanted to know the limiting distribution of M_n…

by michaelnhowes

“Uniformly random”

The term “uniformly random” sounds like a contradiction. How can the word “uniform” be used to describe anything that’s random? Uniformly random actually has a precise meaning, and, in a sense, means “as random as possible.” I’ll explain this with an example about shuffling card. Shuffling cards Suppose I have a deck of ten cards…

by michaelnhowes

The discrete arcsine distribution

The discrete arcsine distribution is an interesting distribution with connections to random walks, Markov chains and the Beta distribution. This post shows that the discrete arcsine distribution is a special case of the beta-binomial distribution.

by michaelnhowes

The sample size required for importance sampling

My last post was about using importance sampling to estimate the volume of high-dimensional ball. The two figures below compare plain Monte Carlo to using importance sampling with a Gaussian proposal. Both plots use M=1,000 samples to estimate v_n, the volume of an n-dimensional ball A friend of mine pointed out that…

by michaelnhowes

Uniformly sampling orthogonal matrices

An n \times n matrix M \in \mathbb{R}^{n \times n} is orthogonal if M^\top M = I. The set of all n \times n orthogonal matrices is a compact group written as O_n. The uniform distribution on O_n is called Haar measure. There are many ways to generate random…

by michaelnhowes

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