Poisson Distribution: Definition and Examples

Introduction:

A probability distribution is a function that gives the relationship between the outcome of a random variable in any random experiment and its probable values. In this article, we will discuss one of the probability distributions which is commonly used in Data Science, Poisson Distribution : Definition and Example.

To know more about Random Variable, read the article Introduction to Probability.

To know about other probability distributions, read the article Probability Distribution used in Data Science.

Table of Content:

• Poisson Distribution
• Conditions for Poisson Distribution
• Mathematical Definition
• Properties of Poisson Distribution
• Relation between Poisson and Binomial Distribution

Poisson Distribution:

Poisson Distribution (named after the French mathematician Denis Simon Poisson) is a discrete probability distribution that measures the probability of a random variable over a specific period of time.

Example: 

• Number of arrivals at a restaurant
• Number of calls per hour in a call center

Conditions for Poisson Distribution:

• An event can occur any number of times in the defined period of time
• All the events are independent
• The rate of occurrence of events is constant

Mathematical Definition:

For any random variable X, the distribution function for Poisson Distribution is given by:

To know more about the mean, read the article on Measures of Central Tendency.

Example: Poisson distribution using Python

1.

# import libraries
 
import matplotlib.pyplot as plt
from scipy.stats import poisson #poisson: poisson distribution function
 
# generating poisson distribution for sample size of 1000
 
sample_set = poisson.rvs(mu = 5, size = 1000) #poisson.rvs : generate the random number
 
# plot the poisson distribution
 
plt.hist(sample_set, edgecolor = 'red')

2. Poisson distribution at the different mean values for the same sample size

# import library
 
from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns
 
# plotting poisson distribution for different mean values
# lam : mean value
sns.distplot(random.poisson(lam=20, size=1000), hist=False, label='mean = 20')
sns.distplot(random.poisson(lam=50, size=1000), hist=False, label='mean = 50')
sns.distplot(random.poisson(lam=80, size=1000), hist=False, label='mean = 80')
 
plt.legend()
plt.show()

From the above figure, we get as the mean value increase the curve become flatter and shorter.

Properties of Poisson Distribution:

• Poisson distribution has only one parameter i.e. mean
• Mean = Variance
• It tends to normal distributions, if mean tends to infinity

To know more about normal distribution, read the article on Normal Distribution: Definition and Example.

Relation between Poisson and Binomial Distribution:

Poisson distribution is a limiting case of Binomial Distribution.

i.e. when we increase the value of n to infinite we get the Poisson distribution.

The distribution function of Binomial Distribution is given by:

To know more about Binomial distribution, read the article Binomial Distribution: Definition and Example.

Now, substituting the value of p in B(x: n, p), we get:

This is the required Poisson distribution function.

Conclusion:

In this article, we have discussed about one of the most important probability distribution Poisson Distribution , with examples in python.

Hope this article will help in your data science and machine learning journey.

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