discrete_distribution_from_uniform.py

import math
import random

def poisson_point_probability(k: int, lambda_: float) -> float:
    return math.e**(-lambda_)*lambda_**k/math.factorial(k)

def accum_poisson_probability_distribution(lambda_: float, cutoff_prob_error = 0.000_000_01) -> list[tuple]:
    F_x = [(0, poisson_point_probability(0, lambda_))]
    while F_x[-1][1] < 1 - cutoff_prob_error:
        F_x.append((len(F_x), F_x[-1][1] + poisson_point_probability(len(F_x), lambda_)))
    F_x[-1] = (F_x[-1][0] - 1, 1)
    return F_x

def single_experiment(accum_probability_distribution: list[tuple]) -> int:
    uniform_random_value = random.random()
    i = 0
    while accum_probability_distribution[i][1] < uniform_random_value:
        i += 1
    return accum_probability_distribution[i][0]

def n_experiments(accum_probability_distribution: list[tuple], n: int = 1) -> list:
    return [single_experiment(accum_probability_distribution) for _ in range(n)]

if __name__ == "__main__":
    lambda_ = 10
    n = 1_000_000
    poisson_distribution = accum_poisson_probability_distribution(lambda_)
    results = n_experiments(poisson_distribution, n)
    expected_value = sum(results)/n
    variance = sum([(value - expected_value)**2 for value in results])/n
    print(expected_value, variance) # 10.008362 10.017930076999944; exec. time (user + sys) 0m1,232s + 0m0,024s