In this article, we’ll explore the characteristics of the Poisson distribution and visualize it using Python. We will utilize the scipy.stats library for generating random variables and the matplotlib library for creating plots. Let’s Dive In !
Introduction
The Poisson distribution is a discrete probability distribution often used to model occurrence of events within a specific interval.
Step 1 — Introduction to the Poisson
The Poisson distribution is a discrete probability distribution similar in shape to the normal distribution but with varying degrees of skewness. It’s commonly used to estimate the number of events occurring within a fixed interval and assuming that these events happen at a constant average rate.
Step 2 — The Poisson Distribution Formula
The Poisson distribution is characterized by the formula:
P(X=k)=k!e−μ⋅μk Where:
- P(X=k) is the probability of observing k events in the interval.
- μ is the average rate of occurrence of the event.
- e is Euler’s number (approximately 2.718...).
- k! represents the factorial of k.
Step 3 — Use Cases of the Poisson Distribution
The Poisson distribution is suitable for modeling independent events occurring at a constant rate within a given time frame. Let’s take an example to understand, if we include the number of users visiting a website within an interval or the count of cars passing a specific road segment in a fixed time.
Step 4 — Probability Calculation Using the Poisson Distribution Formula
To calculate the probability of observing a specific number of events (k) within an interval, we can use the Poisson distribution formula for finding the probability that not more than 10 cars pass a road segment in a minute, we sum up the probabilities for 0, 1, 2, ... 10 cars passing.
Step 5 — Visualizing the Poisson Distribution
We will use the scipy.stats.poisson module to generate random variables from a Poisson distribution with different mean values. Then, we will create histograms to visualize the distributions.
Step 7 — Interpretation of Plots
- The first plot corresponds to a Poisson distribution with a mean (μ) of 2. It exhibits a highly skewed distribution which aligns with the characteristics of a Poisson distribution with a low mean.
- The second plot represents a Poisson distribution with a mean of 10. This distribution is less skewed compared to the first plot and it approaches a more symmetric shape resembling a Gaussian (normal) distribution.
Conclusion:
In this article, we will explored the Poisson distribution and its formula, use cases and how to visualize it using Python. By generating random variables and creating histograms we have demonstrated the impact of changing the mean parameter on the shape of the distribution. The Poisson distribution provides valuable insights into scenarios involving the occurrence of discrete events within fixed intervals.
