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
counts = np.array([1,3,4,5,6,7,8,9,10,11,12,13])
occurences = np.array([1,2,3,6,9,11,8,8,6,2,1,1])
n_expts = np.sum(occurences) # check
print(n_expts)
58
total = occurences@counts # Dot product
print("total # of counts =", total)
total # of counts = 423
This is the best estimate of the mean of the parent Poisson distribution.
mean = total/n_expts
print('mean counts/second =', mean)
mean counts/second = 7.293103448275862
Use the cumulative distribution function for the Poisson distribution
probability1 = stats.poisson.cdf(5, mean)
print("probability of getting 5 or fewer counts =", probability1)
n_expected1 = n_expts*probability1
print("expected number of trials with 5 or fewer counts =", n_expected1)
probability of getting 5 or fewer counts = 0.2648489299548199 expected number of trials with 5 or fewer counts = 15.361237937379554
Use the cumulative distribution function to find the probability of 19 or fewer counts. The probability of 20 or more is (1 - the probability of 19 or fewer).
probability2 = 1 - stats.poisson.cdf(19, mean)
print("probability of getting 20 or more counts =", probability2)
n_expected2 = n_expts*probability2
print("expected number of trials with 20 or more counts =", n_expected2)
probability of getting 20 or more counts = 7.674243593669416e-05 expected number of trials with 20 or more counts = 0.0044510612843282615
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 scipy, matplotlib
| Software | Version |
|---|---|
| Python | 3.11.5 64bit [MSC v.1916 64 bit (AMD64)] |
| IPython | 8.15.0 |
| OS | Windows 10 10.0.26100 SP0 |
| scipy | 1.11.1 |
| matplotlib | 3.7.2 |
| Sat Feb 08 13:57:21 2025 Eastern Standard Time | |