Poisson Probability Distributions

11.3. Poisson Probability Distributions#

Further Reading: §4.3 in Navidi (2015)

11.3.1. Learning Objectives#

After studying this notebook, completing the activities, engaging in class, and reading the book, you should be able to:

  • Model scientific and engineering problems using the Poisson distribution.

import numpy as np
import math
import matplotlib.pyplot as plt

11.3.2. Definition#

The Poisson distribution is important for modeling rare events including failures and radioactive decay.

If \(X \sim\) Poisson(\(\lambda\)), then:

  • \(X\) is a discrete random variable whose possible values are the non-negative integers (\(x\) is unbounded)

  • The parameter \(\lambda\) is a positive constant

  • The probability mass function is

\[\begin{split}p(x) = P(X = x) = \begin{cases} e^{-\lambda} \frac{\lambda^x}{x!} & \text{if } x \text{ is a non-negative integer} \\ 0 & \mathrm{otherwise} \end{cases}\end{split}\]

  • The Poisson probability mass function is very close to the binomial probability mass function when \(n\) is large, \(p\) is small, and \(\lambda = n p\).

  • The mean is \(\mu_X = \lambda\)

  • The variance is \(\sigma_X^2 = \lambda\).

Alright, those are nice properties, but how do you specify \(\lambda\)? Often the Poisson distribution is used to model the number of rare events during a fixed duration. In this context, \(\lambda\) is the expected number of rare events. Let’s look at an example.

11.3.3. Example: Failure Rates#

Consider a flood occurs on a specific river once every 100 years. Calculate the probability 0, 1, 2, …, 6 floods occur in the next 100 years. Assume the Poisson distribution accurately models flooding on this river.

Step 0: Identify \(\lambda\)

We are modeling floods, which are a rare event. The fixed duration is 100 years. We expect 1 flood every 100 years, so \(\lambda = 1\).

Step 1: Write a function to evaluate the PDF

def poisson(k,lmb):
    ''' PMF for Poisson distribution
    Args:
        k: number of events
        lmb: average number of events per interval
    Return:
        probability of k events occuring during interval
    '''
    
    assert k >= 0
    assert lmb >= 0
    
    return np.exp(-lmb) * lmb**k / math.factorial(k)

Step 2: Test out function for 10 floods

poisson(10,1)

1.0137771196302975e-07

Step 3: Answer the question asked

Home Activity

Evaluate the probabilities of 0 to 6 floods during the next 100 years. Store your answer as a numpy array p_floods such that p_floods[3] is the probability of 3 floods.

n_floods = range(0,7)
p_floods = np.zeros(7)

# Add your solution here

# Removed autograder test. You may delete this cell.

Using our results, we can plot the probability mass function and cumulative density function:

plt.bar(n_floods,p_floods)
plt.xlabel("Number of Floods")
plt.ylabel("Probability")
plt.title("Probability Mass Function")
plt.show()
../../_images/4ee92e54403059c28ee94cf88265ee446380a9a7f52fee27d313a0722aee3298.png 
# calculate cumulative sum
cum_p_floods = np.cumsum(p_floods)

# Create the plot
plt.plot(n_floods,cum_p_floods)
plt.xlabel("Number of Floods")
plt.ylabel("Probability of N or fewer floods")
plt.grid(True)
plt.title("Cumulative Distribution Function")
plt.show()
../../_images/7724dd0f88837baab9926eeade699fa0235afa0a9151af000acd1888343ae624.png
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