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The Poisson distribution is a probability distribution that is commonly used to model the occurrence of rare events over a specific interval of time or space. It provides a mathematical framework for estimating the likelihood of a certain number of events happening within a given timeframe. In this guide, we will explore the concept of the Poisson distribution and show you how to calculate probabilities using this powerful statistical tool.

The Background

This distribution comes with a quite impressive historical background. This distribution was established by a mathematician named Simon Dennis Poisson. This distribution became essential when identifying mistakes and challenges that occur in data sets.

When Should I Use The Poisson Distribution?

The Poisson distribution is used to model the number of times a mistake has occurred in a particular data set. For example, if you are a developer, the number of failed app downloads would count towards this distribution. This distribution can be used to determine the likeliness of occurrences per unit. Back to our original example, you could use this distribution to identify challenges per user, per download, or per cell phone carrier. This method is especially useful for analyzing data when there are few trials and the likelihood of failure is high.

Understanding The Poisson Experiment

An experiment containing this methodology will have the following characteristics:

  • The trial produces outcomes that can be considered failures or successes
  • The typical success rate of these occurrences
  • The likelihood of success is appropriate when compared to the region size
  • The likelihood of success is less the smaller the region is

The Poisson Formula

Here is the formula for the Poisson Distribution:

f(x) = P(X=x) = (e-λ λx )/x!

Where:

P(X=x) = Probability of exactly (x) occurrences in an interval

Λ (lambda) = Mean number of occurrences during interval

X = number of occurrences desired

e = base of the natural logarithm

Understanding the elements of this formula is essential before using it to analyze your data sets. It is important to review all of the different symbols prior to starting your trials.

We recommend downloading this distribution table online, to have beside you as you begin analyzing your data.

Visualizing The Poisson Distribution

In order to get to the root of a problem, danger, or challenge is identified in a dataset, you will need to examine the records. The issue typically lies in the number of problems occurring in a specific trial. In this example, we will assume that the average number of problems occurring each week is 5. We will use this data to now calculate the likelihood of:

  • Having more than three accidents occur per week
  • Having less than two accidents occur per week

What Happens When There Are Fewer Than 2 Accidents Per Week Using the Poisson Distribution?

First and foremost, we will need to calculate the probability of having less than two accidents and a specific week. The likelihood of less than two signifies the possibility of no accidents and the second possibility of 1 accident occurring in the week’s time. Using everything you know about the rules of probability, it is now time to customize the formula to set this particular problem.

Your main variable should refer to the number of occurrences that are wanted out of the trial. For example, you can use P (x equals 0) + P (x equals 1). The main components of the formula signify the different factorials of 0 + 1. Keep in mind to always use an exclamation mark when using a 1.

Calculating the Probability of More Than Three Accidents

Next, we will move forward with calculating the likelihood of more than three accidents occurring in a single week. In this model, we will use the distribution P (x is greater than 3). The likelihood of more than 3 indicates the first probability, the 2nd probability of one incident, the third probability of 2, and the fourth probability of 3 incidents. Due to the rules of probability, you must abide by the rules of customization.

Revisiting The Formula

The X variable will correspond to the number of occurrences expected. The first equation will appear as ‘1 – {P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)}’. The focal point component is repeated four times in four separate segments. The first segment is ‘P(X=0)’. The second segment is ‘P(X=1)’. The third segment is ‘P(X=2) and the fourth segment is ‘P(X=3)’.

Understanding the Poisson Distribution

The Poisson distribution is named after French mathematician Siméon Denis Poisson, who introduced it in the early 19th century. It is particularly useful for modeling events that occur randomly and independently, such as the number of customer arrivals at a service counter, the number of defects in a manufacturing process, or the number of phone calls received in a call center during a specific period.

The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of event occurrences within the given interval. It assumes that the events are rare and that the probability of more than one event occurring simultaneously is negligible.

The probability mass function (PMF) of the Poisson distribution is given by the formula:

P(x; λ) = (e-λ * λx) / x!

where:

  • P(x; λ) represents the probability of x events occurring in the interval.
  • e is the base of the natural logarithm, approximately equal to 2.71828.
  • λ is the average rate of event occurrences.
  • x is the number of events.
  • x! denotes the factorial of x.

Applications of the Poisson Distribution

The Poisson distribution has various applications in different fields:

  • Queueing theory: It is used to model the arrival and service rates in queuing systems.
  • Reliability engineering: It helps assess the reliability and failure rates of systems.
  • Insurance risk analysis: It is used to estimate the number of claims within a certain period.
  • Environmental science: It can model the occurrence of rare events, such as earthquakes or floods.

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Conclusion

The Poisson distribution is a valuable tool for calculating probabilities related to rare events. By understanding the concept of the Poisson distribution and following the step-by-step calculation methods outlined in this guide, you can improve your ability to analyze and predict the occurrence of events within a specific interval. Incorporate the Poisson distribution into your statistical analysis toolkit and enhance your data-driven decision-making process.

Have you led projects and are looking to earn a project management certification?  You might be interested in learning about how lucrative this can be.  Check out these articles.

Should You Become PMP Certified? – Here’s Why Becoming A Certified Project Management Professional Can Boost Your Career

13 PMP Benefits Once You Get The PMP Certification

No experience leading projects but still want to get into project management?  No problem!  Check out these articles.  

CAPM Certification Eligibility

What is a Certified Project Manager; How do I get PM Certifications

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The course is amazing at teaching the basics but the exams don’t match up with the PMBOK vocabulary or really the slide content. So many questions on agile project management but not even an entire section dedicated to it.

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Engaging, self-pacing course that contains accurate content. Prepared me well for the PMP exam which I passed on my first attempt. This course is best for those who would like to learn at their own pace. For example, I completed it over the span of 2 and a half months.

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The course was thorough and the practice exams enhanced my learning. However, I am having a hard time receiving my PDU credit for taking the course.

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The Course documents should remain open for longer than 6 months, given that PMI gives you 365 days to prepare for exam. Also, the option to extend visibility for the course documents, costs almost a 5th of the course, but opens up the docs for just 30 more days. Propose to extend this to 90 days at the very least.

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Your course, quizzes at the end of most modules and practice exams truly challenged me. I definitely derived the value and knowledge I was looking for in preparation for taking the PMP Certification Test. In particular, I appreciated the explanations that popped-up in the tests after an answer was submitted - very educational and helpful to my learning/understanding the material and concepts better. Some areas that could be improved (and some of this is nit-picking) include: * Some of the grammar and overall sentence structures, particularly in the latest set of practice exams, needs to be cleaned-up. Some were missing 'of', 'for, or had an extra 'an' in the narrative - got confusing at times. * In the practice exams, I'd like to see shorter sentences rather than the flurry of 'and' and 'or' conjunctions that made reading the sentences go on forever (had to read them multiple times to truly understand their meaning). That said, I enjoyed the overall online course taking experience and thank you for accommodating my requested time extensions. Best regards, Scott Conn

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Liked it. Need access for longer though.

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What I like best about Brain Sensei is the interactive nature of the course. I think this material would be numbing if delivered as simple lectures with slides. This method is an improvement on that. On the other hand, an in-person course, while more expensive, would likely be even more useful if you have the funds and the time. I haven't taken the PMP exam yet, so I can't comment on how useful the practice tests are.

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Passed first attempt

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Truly the best course in town if you wish to pass the PMP exam on the first try. I love the idea of playing the role of the samurai and practice your project management skill in her shoes. And not to forget, the generous amount of practice exam gave me just the right dose of confidence before taking the real one. Highly recommended

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The course was great, though I found a lot of parts hard to comprehend. I feel a lot more prepared and better skilled to manage projects in my career. Thanks, Brain Sensei.

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The BrainSensei PMP preparation course provided me with all the material required to be ready for the exam. It is well structured, explains the concepts clearly, and most importantly, provides ample training questions through the four mock exams and assessment questions at the end of each module. I passed the exam with Above Target in all sections on the first attempt.

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Overall I really liked this learning course. It was very helpful and allowed me to be well prepared for the exam. The 2nd and 3rd practice exam are great to give you real example questions and help prepare you for what to expect. BrainSensei's customer service was the best. I had to delay taking my test due to COVID and they extended my membership for free. They went above and beyond my expectation which was great. Only changes I would recommend is to add 1 or 2 more practice exams like the 2nd and 3rd exam and get rid of the agile practice exam. This is not a real focus of the test and therefore is not really applicable in my opinion.

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BrainSensei developed a great learning method to prepare for the PMP certification exam. Their customer service is always very professional, fast, spot on and dedicated to your success. I recommend them.

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I've sent several emails to BrainSensei (one of them not so nice, but the rest were okay) and the team has always been so polite in answering the questions. I'm a very straight-forward person and don't like the PMI-ism type questions because they seem like 'gotchas' or traps, and this is not how I manage my team. I took the Master of Project course (which was garbage) then failed the PMP exam. Throughout the PMP exam I felt so out of touch. Even with the first practice exam on BrainSensei I felt out of place, but I am not getting the hang of things. I hope to pass the exam on the second attempt! The brain sensei experience has been *miles* ahead of the Master of Project garbage. Brain Sensei should be proud of both their product and of their team! I love getting replies to my concerns so quickly. Nice work :) !!!

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This course was great. it laid a great foundation for the PMP exam.

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very useful

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I was very pleased with the preparation provided for the PMP certification. I went through the materials twice and passed the exam on the first try. A couple of the questions need to be polished up, but very good training overall.

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I have studied with other programs and I didn’t feel prepared. After a short time with Brain Sensei, I had the confidence to succeed on exam day, which I did on the first try after studying with Brain Sensei. It was worth every penny.

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Course is made for people who are already grounded in the material, not for someone that needs all the basics. Was budget friendly and OK content.

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Overall layout is good, but there are some quality issues. Progress isn't always saved across all devices and progress gets lost. Some small spelling mistakes. I loved the practical storyline...

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Passed my PMP on first attempt using Brain Sensei Prep course.

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Quick and helpful response. Thanks!!

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Need improvement in presenting more relevant real exam type questions. The practice exams are too simple and easy and do not prepare you for the real exam. Otherwise, good concept presenting material

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The prep materials provided plenty of content which prepared me well for the PMP exam. The questions were packed with information.

- Cameron

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Teaches the PMP in a solid manner, some things you need to know as a potential consumer: -The Samurai backstory is extremely short, and didn’t really assist me with recall in any in-depth way. It only gave broad brush-strokes as to what each phase entailed. -MOST of what was on the test was covered in the lessons, however if I had zero real prior PM knowledge prior to the test, I would have failed. -The first practice test forces you, as the student to actually read into the PMBOK, which was a benefit in the long run as it assisted me with quickly referencing what I needed to find. -The course will likely cover wave tops and subjects, forcing me to self-study for a week before my test... not a bad deal all things considered. -Brain Sensei’s course has incorporated Agile into one of their practice tests, and this test IS mandatory. This may seem strange, and I didn’t get a satisfactory reason why I needed to take a purely Agile test when I only wanted to take the PMP, for anyone else in this scenario, this is the reason: In June, PMI is revamping the PMP test to include Agile as a significant portion of the PMP. Brain Sensei is just ahead of the curve trying to get students prepared... fret not. Their customer service is extremely responsive and does take care of their members. My program had a couple bugs, and the customer service fixed my problems within 24 hours. All in all... it was absolutely worth the money for me.

- Kevin

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I am very impressed with this learning system. The content is laid out in a way that will make any learner understand all the process groups and ITTOs in order to understand Project management and be equipped to excel in the test. The learning system is engaging and fun at the same time. It's very different than other learning material which is composed mostly of videos. Some improvements are needed, but in my opinion Brain Sensei is great value for money

- said durra

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At first, I was attracted to Brain Sensei training for their original learning program using a Samurai's story. And indeed it made the whole learning experience funnier - so easier. I felt like the Samurai. Then, I was amazed by The Brain Sensei Team's reactivity and help when I reached out through contact@brainsensei.com Their support was constantly reliable and efficient. I appreciated their professionalism and didn't feel on my own. At last, the course and the practice exams proved to be efficient since I passed the PMP exam - with 'Above Targe't in all Domains. So thank you so much Brain Sensei Team !! Charlene

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Overall very good. Got stuck a few times trying to find resources which I should have downloaded the before starting the program. Customer service has been great.

- Jonathan Williams

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SO far I am thoroughly enjoying the Brain Sensei PMP prep course. I have taken a Project management class as a part of my degree which I recently completed and project management has always interested me. I find the course to be educational and easy to follow. I am grateful that this course has been designed in such a way that is simple to understand all of the complexities in project management and that allows me to study at my own pace.

- Eric Browne

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The content is good and easy to understand

- Glenn

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This review is based on completing all course work through module 102. The story and interactive nature of the course are helpful for learning and relating the material. It would be nice to have the page numbers to the PMBOK guide where references are made to it. Other information that will be helpful, are the explanations to the practice questions at the end of each module. These are provided only after immediately answering the questions. Having these available in the score summary from the 'history' tab will be helpful to refresh why the answer is correct and other options are not.

- Rohan Punit

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Overall I would rate this course an 8/10. I liked how there were exams towards the end of every module and I felt that it was very useful having it laid out in different scenarios. I rate it only an 8/10 as I had some glitch/software issues and lost my progress resulting in me having to re-do modules/exams multiple times. All in all, that was maybe useful in the end as it was a way for me study or 're-learn' the course material. Thank you!

- Nikki Cordy

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Clear and precise explanations of how things work. I was never lost navigating the course.

- Gary Nicol

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A smart approach to present the information which makes it fun and engaging while at the same time preparing for the PMP exam. Brain Sensei differentiates itself from other PMP exam preps in that it delivers the knowledge through the use of storytelling technique. I recommend this course because it will help you pass your PMP exam on the first try.

- Jamal

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Loved the story telling and the practice exams. I felt very prepared when the actual exam came around.

- Josh

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One of a kind training that was truly engaging!

- Claire

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Excellent course for preparing for the PMP exam in a fun and engaging way! The practice questions resemble the actual questions on the exam making it a completely worthwhile experience.

- Rossetta

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I passed the PMP exam after taking Brain Sensei's Online PMP Certification Exam Prep. The practice exam questions were so similar to the questions that appeared on the exam. On exam day, when I sat down to take the test and started running through the questions I felt so relieved! The practice exams are also timed so you get more comfortable with the exam experience and feel better prepared when you sit down to take the test. The other differentiator was the video content throughout the modules. The concepts are taught through a story of a young samurai defending her kingdom. This will probably be the only PMP prep course where you can learn all the knowledge areas and feudal Japan at the same time. I'd recommend this course not just because of the videos but because it helped me pass my exam on the first try.

- Jackie

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Good course, but a few grammatical errors with how some questions were worded.

- Sonia

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It was good!

- Sergei

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Loved the unique approach and it still had all the info I'd expect from a full PMP exam prep course. Very responsive on questions too.

- Chris