NOTE: In this notebook I use the stats sub-module of scipy for all statistics functions, including generation of random numbers. There are other modules with some overlapping functionality, e.g., the regular python random module, and the scipy.random module, but I do not use them here. The stats sub-module includes tools for a large number of distributions, it includes a large and growing set of statistical functions, and there is a unified class structure. (And namespace issues are minimized.) See https://docs.scipy.org/doc/scipy/reference/stats.html.
import numpy as np
from scipy import stats
If we use the minute as our unit of time, the determination of the mean count rate is trivial: it's $R = 270\, \mbox{min$^{−1}$}$. If we choose seconds, it's only slightly less trivial: $R=270/60 = 4.5\, \mbox{s$^{−1}$}$.
We are sampling from a Poisson distribution, so the error given, by the standard deviation, is $\sigma = \sqrt{\bar \mu}$:
sigma = np.sqrt(270)
print('sigma (min^-1) =', sigma, '; sigma (s^-1) = ', sigma/60)
sigma (min^-1) = 16.431676725154983 ; sigma (s^-1) = 0.27386127875258304
print('fractional error =', sigma/270)
fractional error = 0.06085806194501846
n_expected = 15*270
print('expected in 15 minutes =', n_expected)
expected in 15 minutes = 4050
The probability of actually getting this number of counts is given by the probability mass function of the Poisson distribution:
probability = stats.poisson.pmf(n_expected,n_expected)
print('probability of getting exactly', n_expected,' counts =', probability)
probability of getting exactly 4050 counts = 0.006268644164779241
version_information is from J.R. Johansson (jrjohansson at gmail.com); see Introduction to scientific computing with Python . If not already installed on your machine, run pip install --upgrade version_information from the terminal
%load_ext version_information
version_information numpy, scipy
| Software | Version |
|---|---|
| Python | 3.11.5 64bit [MSC v.1916 64 bit (AMD64)] |
| IPython | 8.15.0 |
| OS | Windows 10 10.0.26100 SP0 |
| numpy | 1.23.2 |
| scipy | 1.11.1 |
| Sat Feb 08 13:58:14 2025 Eastern Standard Time | |