Monte Carlo estimations of e

Question

I need to estimate $e$ with a monte carlo method. We only learned the crude monte carlo integration, so I can't use any robust monte carlo simulations.

I know that $\displaystyle \int\limits_1^x \frac{1}{s}ds=\ln(x),$ so I just need to play around with the limit using randomly generated numbers.

You would normally use MC to integrate something yielding $e$ in the end, no? — gt6989b, Jan 28 '14 at 18:46
You may want to check http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_Uniform_E_EstimateExperiment — Macavity, Jan 28 '14 at 19:06
A totally different idea: The probability that a random permutation of $n\gg 0$ objects is fixed-point free, is $\approx \frac1e$ — Hagen von Eitzen, Jan 28 '14 at 19:54

score 4 · Answer 1 · answered Jan 28 '14 at 18:46

The solution below assumes that we know how to take $r$-th roots, where $r$ is rational. I do not consider it a good solution.

Use the obvious Monte Carlo estimation of $\int_1^3 \frac{dx}{x}$ to estimate $\ln 3$. Call this estimate $r$.

Then $\ln 3\approx r$, and therefore $3\approx e^r$. Now calculate $3^{1/r}$.

Monte Carlo estimations of e

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