# Repeated estimates from random lifetimes

Run `python reproduce.py` with Python, numpy and matplotlib. This reproduces a seeded computational experiment, not a physical measurement. The assigned population median is 185.52 seconds, a teaching timescale with no isotope identity claim.

Each lifetime is exponential with rate ln(2)/185.52 per second. A single Python random.Random(21001) stream is consumed in this exact order: N=100, then1,000, then10,000;200 repetitions per N, and N complete lifetimes per repetition. statistics.median averages the two central order statistics for these even sample sizes. The generator reproduces the original random-number order; it does not infer missing samples from summary statistics.

median-estimates.csv contains600 reproduced estimates; summaries.csv contains three summaries. Relative standard deviation is the sample standard deviation with denominator199, divided by the assigned population median (not the estimated mean). Values are0.14279394027719466,0.04614893160261974 and0.014471117215130993. All stored historical summary fields match exactly on this runtime. Original prototype/results are read only and never overwritten. If the historical file is absent in a standalone download, reproduction still runs and the optional historical comparison is null, not a claim of verification.

For independent complete observations, the asymptotic relative standard deviation of the sample median is1/[ln(2)sqrt(N)]. This follows from Var(median) approximately1/[4N f(median)^2], where the exponential density at its median is lambda/2. It is not a universal clock limit, count-rate error or a finite-sample confidence interval. asymptotic-curve.csv has101 distinct integer sample sizes on a logarithmic grid100–10,000.

survival.csv has501 times spanning zero through five assigned median times, with expected surviving fraction exp(-lambda*t). It describes the population expectation, not a simulated finite count, nor the total mass of parent and products. A half-life does not produce a periodic oscillator.

Figures: median-spread shows all600 estimates as boxplots (quartiles, median,1.5IQR whiskers and all outliers), plus empirical standard deviations against an asymptotic curve. Boxes are not confidence intervals. expected-survival shows the analytic expectation. PNG, SVG and PDF are provided.

Independent checks recompute standard deviations through explicit sums, verify exact historical summary parity, survival at0/one/five median times, monotonic survival and inverse-square-root scaling. Detector efficiency, censoring, dead time, background, correlations and systematic errors are excluded. Total CSV rows1,205.

Sources: [NIST exponential model](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm), which gives exponential survival and median; [BIPM second definition](https://www.bipm.org/en/si-base-units/second), which concerns a fixed atomic transition frequency rather than spontaneous-decay timing. The median-uncertainty expression is derived for the stated statistical model.