# The answer is e, what was the question?!

**R – Xi'an's Og**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**A** rather exotic question on X validated: *since π can be approximated by random sampling over a unit square, is there an equivalent for approximating e?* This is an interesting question, as, indeed, why not focus on *e* rather than* π *after all?! But very quickly the very artificiality of the problem comes back to hit one in one’s face… With no restriction, it is straightforward to think of a Monte Carlo average that converges to *e* as the number of simulations grows to infinity. However, such methods like Poisson and normal simulations require some complex functions like sine, cosine, or exponential… But then someone came up with a connection to the great Russian probabilist Gnedenko, who gave as an exercise that the average number of uniforms one needs to add to exceed 1 is exactly *e*, because it writes as

(The result was later detailed in the American Statistician as an introductory simulation exercise akin to Buffon’s needle.) This is a brilliant solution as it does not involve anything but a standard uniform generator. I do not think it relates in any close way to the generation from a Poisson process with parameter λ=1 where the probability to exceed one in one step is *e*⁻¹, hence deriving a Geometric variable from this process leads to an unbiased estimator of *e* as well. As an aside, W. Huber proposed the following elegantly concise line of R code to implement an approximation of *e*:

`1/mean(n*diff(sort(runif(n+1))) > 1)`

Hard to beat, isn’t it?! (Although it is more exactly a Monte Carlo approximation of

which adds a further level of approximation to the solution….)

Filed under: Books, R, Statistics Tagged: Buffon’s needle, cross validated, Gnedenko, Monte Carlo integration, Poisson process, simulation

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