A rate parameter, written λ (lambda), gives the average number of events expected within a specified time or space interval. In a Poisson distribution, λ equals both the mean and the variance, and the probability mass function uses it to estimate the chance of observing k events. section 3.1.4
Use λ to interpret a Poisson model:
- Treat λ as the average event rate for a fixed interval, such as hourly calls, daily absences, or annual accidents. section 3.1.4
- Use historical event frequencies to estimate future probabilities while assuming a constant average rate and independent events. section 9.3.1
- Expect smaller counts to dominate at low λ; as λ increases, the distribution becomes more symmetric and approaches a normal shape. section 3.1.4
Call-center absenteeism
With λ = 2 absences per day, the probability of zero absences is 13.53%, while one and two absences each have probability 27.07%. These probabilities support staffing decisions and can seed a Monte Carlo simulation. section 9.3.1 section 9.4
>>> import numpy as np #1
>>> import pandas as pd #2
>>> from scipy.stats import poisson #3
>>> lam = 2 #4
>>> k_values = np.arange(0, 9) #5
>>> probabilities = poisson.pmf(k_values, lam) #6
>>> probabilities_percentage = \
>>> np.round(probabilities * 100, 2) #7
>>> poisson_table = pd.DataFrame({'k': k_values, \
>>> 'Probability (%)': probabilities_percentage}) #8
>>> print(poisson_table) #9
k Probability (%)
0 0 13.53
1 1 27.07
2 2 27.07
3 3 18.04
4 4 9.02
5 5 3.61
6 6 1.20
7 7 0.34
8 8 0.09
Inspect a Python table that calculates Poisson probabilities for k = 0 through 8 with λ = 2 using SciPy, NumPy, and pandas. section 9.3.1
