Poisson Distribution in Data Science

Poisson Distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space given a constant average rate of occurrence. Unlike the Binomial Distribution which is used when the number of trials is fixed, the Poisson Distribution is used for events that occur continuously or randomly over time or space. This makes it suitable for modeling rare events like accidents, phone calls or website hits. The distribution is defined by its mean λ which represents the expected number of events in the given interval.

Key Concepts of Poisson Distribution

1. Events: Poisson Distribution models the occurrence of events within a given time frame or spatial area. These events must occur independently which means the occurrence of one event doesn’t affect the occurrence of others. Additionally, the events should happen at a constant average rate over the interval.

2. Average Rate (λ): The average rate λ also known as the rate parameter which represents the average number of occurrences of an event in the given time period or spatial area. This value remains constant throughout the observed interval. The parameter λ is central to the Poisson Distribution and finds the shape of the distribution.

3. Time or Space Interval: The interval during which we observe the occurrences of events is important in the Poisson Distribution. This interval can be defined in terms of time (e.g hours, days), space (e.g square miles) or any other metric where occurrences are spread out randomly and independently.

Poisson Distribution Formula

The Poisson Distribution calculates the probability of observing exactly x events in a fixed interval. The formula for the Poisson Probability Mass Function (PMF) is:

P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}

Where:

• P(X=x) is the probability of observing exactly x events in the interval.
• λ is the average rate of occurrences (mean) in the interval.
• x is the number of events for which we are calculating the probability.
• e is Euler’s number which is approximately equal to 2.718.
• This formula allows us to calculate the likelihood of a specific number of events occurring in the given time or space interval assuming that the events occur independently and at a constant rate.

Probability Mass Function (PMF)

The Poisson PMF is used to calculate the probability of exactly x events occurring in a fixed interval. The formula gives us the likelihood of observing x events given the average rate λ.

Example: Call Center

Let’s us consider a call center which receives on average 3 calls per hour (λ = 3) and we want to know the probability of receiving exactly 4 calls in one hour (x=4).

We use the Poisson PMF formula:

P(X = 4) = \frac{e^{-3} 3^4}{4!} = \frac{e^{-3} 81}{24} \approx 0.168

This means that the probability of receiving exactly 4 calls in one hour is approximately 0.168 or 16.8%. By calculating different values of x we can understand the distribution of events for various outcomes.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) of the Poisson Distribution gives the probability of observing at most x events within a fixed interval. It’s the sum of the probabilities from P(X = 0) to P(X=x) which provides the cumulative probability.

The CDF is defined as:

F(x) = P(X \leq x) = \sum_{k=0}^{x} P(X = k)

Example: If we want to know the probability of receiving 3 or fewer calls in one hour we would calculate the CDF as:

P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

This sum gives us the probability of receiving 0, 1, 2 or 3 calls in an hour which is helpful in scenarios where the exact number of events is not important but the total number of events up to a certain point is.

Expected Value of the Poisson Distribution

The expected value (mean) of a Poisson Distribution represents the average number of events we expect to occur in the given time or space interval. For the Poisson Distribution, the expected value is simply:

E[X] = \lambda

For example, if the average number of calls received by a call center is 4 per hour (λ=4), the expected number of calls in one hour is: E[X] = 4

This means we expect to receive 4 calls on average every hour.

Variance and Standard Deviation

1. Variance: The variance of the Poisson Distribution is equal to λ, the average rate of events in the interval. The variance tells us how much the actual number of events deviates from the expected number of events.

\text{Var}[X] = \lambda

2. Standard Deviation: The standard deviation is the square root of the variance which gives us a measure of how spread out the number of events is from the expected value:

\sigma = \sqrt{\lambda}

For example if λ=4, the standard deviation would be: \sigma = \sqrt{4}= 2

Example: Traffic Accidents

Let’s apply the Poisson Distribution in a real-life scenario. Suppose that traffic accidents occur on a certain road at an average rate of 2 accidents per month (λ=2). We can use the Poisson Distribution to calculate the probability of having exactly 3 accidents in a given month. Using the Poisson PMF formula, we get:

P(X = 3) = \frac{e^{-2} 2^3}{3!} = \frac{e^{-2} 8}{6} \approx 0.180

Thus the probability of having exactly 3 accidents in one month is 0.180 or 18%.

Python Implementation for Poisson Distribution

Now let's implement the Poisson Distribution in Python. Here we will be using Numpy, Matplotlib and Scipy libraries for this.

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import poisson

lambda_val = 3  
k = np.arange(0, 10)  
pmf = poisson.pmf(k, lambda_val)

plt.figure(figsize=(8, 6))
plt.bar(k, pmf, color='lightgreen', edgecolor='black')
plt.title('Poisson Distribution PMF (λ=3)', fontsize=14)
plt.xlabel('Number of events (k)', fontsize=12)
plt.ylabel('Probability', fontsize=12)
plt.grid(axis='y', linestyle='--', alpha=0.7)
plt.show()

cdf = poisson.cdf(k, lambda_val)

plt.figure(figsize=(8, 6))
plt.plot(k, cdf, color='purple', marker='o', linestyle='-', linewidth=2)
plt.title('Poisson Distribution CDF (λ=3)', fontsize=14)
plt.xlabel('Number of events (k)', fontsize=12)
plt.ylabel('Cumulative Probability', fontsize=12)
plt.grid(True)
plt.show()

probability_4_events = poisson.pmf(4, lambda_val)
print(f'Probability of exactly 4 events: {probability_4_events:.4f}')

Output:

Probability of exactly 4 events: 0.1680

Relation between Poisson and Exponential Distributions

Poisson Distribution and Exponential Distribution are closely related probability distributions that describe different aspects of the same random process known as the Poisson process. In a Poisson process, events occur randomly and independently at a constant average rate over time or space. These two distributions are conceptually different but share a fundamental connection:

• Poisson Distribution: Models the number of events occurring in a fixed interval of time or space.
• Exponential Distribution: Models the time between consecutive events in the same process.

Both distributions are defined by the same rate parameter λ which represents the average number of events per unit of time or space. The relationship between the Poisson and Exponential distributions can be described as follows:

1. Poisson Distribution is used to calculate the probability of observing a certain number of events (k) in a fixed interval and its formula is:

P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0, 1, 2, \dots

2. Exponential Distribution describes the waiting time between two consecutive events in a Poisson process. Its formula is:

f(x) = \lambda e^{-\lambda x}, \quad x \geq 0

Where:

• λ is the rate parameter, the average rate of events per unit of time.
• x is the waiting time between two consecutive events.

Applications of the Poisson Distribution

Poisson Distribution is used in many real-world scenarios where events occur independently and at a constant average rate:

1. Traffic and Accident Analysis: Used to model the number of accidents occurring at an intersection over a fixed period.
2. Telecommunications: Models the number of calls received by a call center or the number of network requests in a given time period.
3. Medical Field: In healthcare it models rare events like the number of new cases of a disease in a given time period.
4. Queuing Theory: Applied to understand the number of customers arriving at a service point (e.g bank or checkout line) within a certain time period.

By understanding the Poisson Distribution we get valuable insights into modeling rare events over time or space which increases our ability to make informed decisions across various industries.