Table of contents
1.
Introduction
2.
Mathematical Definition
3.
Sample Code
3.1.
Output
4.
Applications of Poisson Distribution
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Poisson Distribution

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Introduction

A Poisson distribution is a distribution that shows the likely number of times that an event will occur within a pre-determined period of time. It is used for independent events which occur at a constant rate within a given interval of time. 

Poisson distribution is 

  • Discrete
  • Used for count-based data

The Poisson distribution, named after the French mathematician Denis Simon Poisson, is a discrete distribution function describing the probability that an event will occur a certain number of times in a fixed time (or space) interval. For example, the number of mails in your mailbox arriving equals the number of customers arriving at a shop.

Mathematical Definition

A Poisson distribution is a distribution that shows the likely number of events occurring in a specific period of time. It is used for independent events which occur at a constant rate within a given interval of time. The Poisson distribution is a discrete function, meaning that the event can only be measured as occurring or not as occurring, meaning the variable can only be measured in whole numbers.

Let’s consider an example, Hema is recording birds in a national park, using a microphone placed in a tree. She is counting the number of times a bird is recorded singing and wants to model the number of birds singing in a minute. For this task, she’ll assume the independence of the detected birds. 

We use the seaborn python library which has in-built functions to create such probability distribution graphs. Also, the scipy package helps in creating the binomial distribution.

The Poisson distribution is denoted as 

Poi is expressed as follows:

for 
k=0,1,2,…
K = events given the parameter 
λ = corresponds to the expected number of occurrences in that time slot.

Sample Code

import numpy as np

import matplotlib.pyplot as plt

import scipy.stats as stats

# n = number of events, lambd = expected number of events 

# which can take place in a period

for lambd in range(2, 12, 2):

    n = np.arange(0, 9)

    poisson = stats.poisson.pmf(n, lambd)

    plt.plot(n, poisson, '-o', label="λ = {:f}".format(lambd))

    plt.xlabel('Number of Events', fontsize=12)

    plt.ylabel('Probability', fontsize=12)

    plt.title("Poisson Distribution varying λ")

    plt.legend()

    plt.savefig('name.png')

Output

  • In each of the different cases, the number assigned to λ corresponds to the peak of the distribution, which then trails off moving further away from the peak.
  • The more events that are expected to take place during the simulation, the greater the expected area under the distribution curve will be.

Applications of Poisson Distribution

  • Calls per Hour at a Call Center
  • Number of Arrivals at a Restaurant
  • Number of Website Visitors per Hour
  • Number of Bankruptcies Filed per Month
  • Number of Network Failures per Week

FAQs

  1. What is the Poisson distribution in statistics?
    In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period.
     
  2. What is Poisson’s law of distribution?
    The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period.
     
  3. What is lambda in Poisson distribution?
    The exponential distribution describes the time between independent events which occur continuously at a constant average rate.

Key Takeaways

In this blog, we discussed:

  • Introduction to Poisson Distribution
  • Mathematical Definition
  • Sample code
  • Applications of Poisson Distribution

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