Over the years I've been volunteering at ProbabilityManagement.org, a not-for-profit that Dr. Sam Savage, author of The Flaw of Averages, started. Their mission is to cure the flaw of averages. The flaw of average states, plans made from average assumptions are wrong on average. Through that organization I learned there has been no closed-form analytical expression for the sum of lognormals.

Why would anybody want the sum of lognormals?

Lognormal distributions, particularly identically distributed ones, are widely employed in various applications due to their unique properties. In finance, they model stock prices or investment returns. In environmental science, they characterize pollutant concentrations or particle sizes, capturing the inherent variability in natural systems. In insurance and reliability analysis, lognormal models assess claim sizes and component lifetimes, reflecting the disproportionate impact of rare but large events. In communication networks, lognormal distributions represent signal strengths or transmission times, acknowledging the potential for interference or propagation delays. Lastly, in biology and medicine, these distributions gauge species abundance, metabolic rates, or drug dissolution rates, capturing the underlying complexity and heterogeneity of living systems. <- written by AI with a few edits.

A practical solution

Tom Keelin, Lonnie Chrisman and Sam Savage recently wrote a paper that outlines a solution. Here's the abstract from the paper:

"The metalog probability distributions can represent virtually any continuous shape with a single family of equations, making them far more flexible for representing data than the Pearson and other distributions. Moreover, the metalogs are easy to parameterize with data without non-linear parameter estimation, have simple closed-form equations, and offer a choice of boundedness. Their closed-form quantile functions (F-1) enable fast and convenient simulation. The previously unsolved problem of a closed-form analytical expression for the sum of lognormals is one application. Uses include simulating total impact of an uncertain number N of risk events (each with iid [independent, identically distributed] individual lognormal impact), noise in wireless communications networks and many others. Beyond sums of lognormals, the approach may be directly applied to represent and subsequently simulate sums of iid variables from virtually any continuous distribution, and, more broadly, to products, extreme values, or other many-to-one change of iid or correlated variables."

I hacked together a javascript version of summing IIDs using an interpolatable (is that a word??) data table. The authors were kind enough to produced an example spreadsheet and that was my my source. Here's the github repo and a codepen which is largely based on their spreadsheet example.

If this is an area of interest for you and you'd like to help, I'm thinking about moving the table into ethfs.xyz and creating a contract with a view functions so anybody can sum independent, identically distributed lognormals. If you're curious and want to learn more about metalog distributions and how to use them hmu on your favorite Farcaster app. @kmacbeth