poisson distribution

Contributed by: Shreya Shetty
LinkedIn Profile: https://www.linkedin.com/in/shreya-shetty-9a070792/

If you’re looking for an easy interpretation of Poisson Distribution, you have come to the right place. Before we get into the details, this is not another formula-based article. When I started my journey in statistics, I wanted every derivation to have a relatable story behind it. I always wondered if I could find a real-life example, or if I should try hard to remember the technical details?  

I began with my adventure of questions and trying to find answers for every topic I had in mind. 

Do you think this is relatable? Read further to keep learning. 

What makes statistics so hard to grasp? Why do we end up losing focus in the end?

  • Set of Derivations
  • Complex Equations
  • Theorems

Instead, if we could get a simple example of where it is being applied, it would be easier to understand. The concept of probability is inseparable in the world of prediction, and so are the distributions.

In probability theory and statistics, a probability distribution is a mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

Simply put, I am rushing to catch a flight and I might miss the flight or I successfully onboard it. Even in the case of cancellation or flight delay, the outcomes remain the same. Either you onboard the flight, or you don’t. 

I hope you have understood the reason for the probability distribution. It is used to be informed of all possible results. Let’s see the types:

  • Discrete: All those similar to the above example.
  • Continuous: Every month we receive the electricity bill and we know it can take any value depending on the usage.

With this background, we can now understand the main topic. Poisson Distribution. 

Also read: Inferential Statistics

What is Poisson Distribution?

A probability distribution that is discrete and has a period associated with it. It is used for independent events that occur at a constant rate within a given interval of time. 

Continuing the flight example, if I need to answer how many people will be on the first flight in the morning. There is no limit to the possible outcomes, isn’t it? To summarise the points:

  • The Poisson distribution is a special case of the binomial distribution (flight onboarded or not). When studying large numbers with a rare (not zero) but the constant occurrence of “successes.”
  • Poisson distribution can be any number of events during the specific period.
  • The distribution is independent and random occurrences of events.
  • The Poisson parameter Lambda (λ) is the total number of events (k) divided by the number of units (n) in the data (λ = k/n), in simplified terms the mean/average for a given period.

Say the toll booth has to determine the maximum traffic time and make sure to avoid congestion. Select a period (morning 10:00 AM to 12:00 PM, usually peak traffic hours), and observe on average how many vehicles pass. Yes, you got the Lambda (λ). Now, use this to answer the probability of a specific vehicle count at the toll booth.

On an average 100 Vehicles arrive, how shall I determine if today is going to be bad, and 200(x) vehicles arrive? Poisson Distribution helps in determining the probability. 

P (x) = λx e-λ

              x!

where, e = 2.172 

Why do we need Poisson Distribution?

A French mathematician Poisson proposed this idea with the example of modeling the number of soldiers accidentally injured or killed from kicks by horses.

It helps model events that are rare and uncommon.

  • Sudden surge of patients to ICU. What is the probability of arrival of more than the expected number of high-risk patients?
  • More than 10 Customers use ATMs during late hours. What is the probability of more than 10 Customers visiting the ATM between 11:30 PM to 11:45 PM.?

It provides us an advantage to prepare a backup plan in case of such unusual events if we know the probability.

In nutshell, it helps:

  • To find the probable maximum and minimum number of times the event will occur within the specified time frame.
  • To find how much variation is likely to be expected from that average number of occurrences.

Where do we apply Poisson Distribution?

Applications are vastly spread and applied in every industry since it addresses the enhancing of efficiency of the operations. Create back-up plans and avoid last-minute surprises.

  • Retail: Inventory maintenance basis the probability predicted.
  • Healthcare: Manage the workforce basis the probability.
  • Manufacturing: Keeping the raw materials ready and avoiding wastage.
  • Telecom: By knowing the peak time, plan for the additional bandwidth to accommodate the smooth connectivity.

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