Overview
The poisson distribution measures the probability of a number of events ocurring. As such, it is a probability over a discrete non-negative variable.Definition
The probability distribution is given by the following
{% P(x) = \lambda^x e^{-\lambda} / x! %}
where {% x %} is an non-negative integer, that is
{% x %}=0,1,2,3.....
This is sometimes re-written in exponential form as
{% P(x) = exp[x log(\lambda) - \lambda - log \Gamma(x+1)] %}
Note, some authors use {% \mu %} in place of {% \lambda %} in the formulas above.
Moments
The moments of the distribution are given by
{% \mathbb{E}[x] = \lambda %}
{% \sigma^2 = \lambda %}
{% \sigma^2 = \lambda %}
Probability of No Events
The probability that no events have ocurred up to time {% t %} is given by
{% P = exp(- \lambda t) %}
Sum of Poisson Variables
Given two Poisson distributions, {% L_1 \sim Pois(\lambda_1) %} and {% L_2 \sim Pois(\lambda_2) %} we have
{% L_1 + L_2 \sim Pois(\lambda_1 + \lambda_2) %}
Simulating a Poisson Random Number
from scipy.stats import poisson
# Set the expected average rate (mu / lambda) and the number of samples (size)
mu_value = 4.0
sample_size = 10
# Generate random numbers from the Poisson distribution
random_numbers = poisson.rvs(mu=mu_value, size=sample_size)
print(f"Poisson random values: {random_numbers}")