**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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**About the Author**

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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