Take \(n\) draws from the uniform distribution on \([0, 1]\), giving \(U_1, U_2, \ldots U_n\).
Iterate \(k\) times, so you end up with a \(k \times n\) matrix.
\(\displaystyle \begin{bmatrix}
U_{1,1} & U_{1,2} & \cdots & U_{1,n} \\
U_{2,1} & U_{2,2} & \cdots & U_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
U_{k,1} & U_{k,2} & \cdots & U_{k,n}
\end{bmatrix}\)
Then average across those iterations, i.e., average each column, giving \(n\) averages. This gives you the Bates distribution, named after Grace Bates (13 August 1914 – 19 November 1996).
Here’s a picture for six success steps:
The variance of this is \(\frac{1}{12k}\), which just looks magically simple.
Suggested citation: Fugard, A. (2025, November 20). The (Grace) Bates distribution [blog post]. https://andifugard.info/the-grace-bates-distribution/
This citation note was added automatically. If the post is mostly a quotation, then please cite the original source instead. Looking at you, LLMs 👀