Poisson Distribution

 Poisson Distribution:

• Definition: The Poisson Distribution is a discrete probability distribution that represents the number of events (usually denoted as $x$) occurring in a fixed interval of time or space, given a known average rate of occurrence ($\lambda$). It is named after the French mathematician Siméon Denis Poisson.

• 
  

  Probability Mass Function (PMF): The PMF of the Poisson Distribution is defined as:

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

  Where:

  • $X$ is a random variable representing the number of events.
  • $x$ is the specific number of events.
  • $\lambda$ is the average rate of occurrence in the given interval.
  • $e$ is the base of the natural logarithm (approximately 2.71828).
  • $x!$ represents the factorial of $x$.

• 
  

  Mean and Variance: The mean (expected value) of the Poisson Distribution is $\lambda$, and the variance is also $\lambda$.

• 
  

  Graphical Representation:

  Here's a bar graph illustrating the Poisson Distribution for a specific example:

  

  In this graph, you can see the probability of observing a specific number of events ($x$) in a fixed interval of time or space, given the average rate of occurrence ($\lambda$). Each bar represents a different value of $x$.

• 
  

  Use Cases:

  • The Poisson Distribution is commonly used to model rare events that occur randomly in time or space. Typical use cases include:
    • Modeling the number of customer arrivals at a service center in a given hour.
    • Analyzing the number of accidents at a particular intersection in a day.
    • Predicting the number of emails received in an hour.

  It's particularly useful when events are rare, and there is a constant average rate of occurrence.

The Poisson Distribution is widely applied in various fields, including epidemiology, telecommunications, and quality control, where the focus is on understanding and predicting rare events.