Quick Python demonstration of Uniform sampling problem (https://twitter.com/fermatslibrary/status/924263998589145090)

number_samples_from_Uniform.py

#!/usr/bin/env python
"""
Problem Statement: Sample from the Uniform distribution over the range [0,1]
until the sum of the numbers sampled is greater than 1. On average, how
many samples are taken?
Answer: e
Problem Source: https://twitter.com/fermatslibrary/status/924263998589145090
Author: Matthew Feickert
Date: 2016-10-28
"""

import sys
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
import itertools  # for fast looping


def simulate_sampling(n_trials):
    """
    Simulate experiments

    Args:
        n_trials: `int` The number of trials

    Returns:
        sum(counts): `int`
        counts: `array of ints`
    """
    counts = []

    for _ in itertools.repeat(None, n_trials):
        n_samples = 0
        sum_ = 0

        while sum_ <= 1:
            sum_ += np.random.uniform(0, 1)
            n_samples += 1
        counts.append(n_samples)

    return sum(counts), counts


def main(n_trials=100000):

    n_samples, counts = simulate_sampling(n_trials)
    result = n_samples / n_trials

    print('Average number of samples taken over {} trials: {}'.format(
        n_trials, result))
    print('Difference between result and e (~{:.6f}): {:.6f}'.format(
        np.exp(1),
        np.absolute(result - np.exp(1))))
    print('Relative difference between result and e: {:.6f}'.format(
        np.absolute(1 - (result / np.exp(1)))))

    # Plot results
    x = list(range(2, 10))
    relative_counts = list(counts.count(number) for number in x)

    plt.plot(x, relative_counts, 'ro', linewidth=1,
             color='black', markerfacecolor='blue')
    line_result = plt.axvline(
        x=result, color='black', label='mean number of samples: {}'.format(result))
    plt.xlabel('Number of samples')
    plt.ylabel('Relative count')
    # Legend
    handles = [line_result]
    handles.append(mpatches.Patch(
        color='none', label='e ~ {:.6f}'.format(np.exp(1))))
    plt.legend(handles=handles)
    plt.savefig('sample_uniform.png')

if __name__ == '__main__':
    if len(sys.argv) > 1:
        main(int(sys.argv[1]))
    else:
        main()

sample_uniform.png