Poisson Distribution & Poisson Process Explained [With Examples]

Poisson distribution is a topic under probability theory and statistics popularly used by businesses and in the trade market. It is used to predict the amount of variation from a given average rate of occurrence within a time frame. This is explained in detail in the following sections.

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Poisson Process

The Poisson process is a widely used stochastic process for modelling the series of discrete events that occur when the average of the events is known, but the events happen at random. Since the events are happening at random, they could occur one after the other, or it could be a long time between two events.

The average time of events is only constant. So, for example, if it is known that in a particular city, an earthquake strikes four times a year on average; this could mean that four earthquakes could occur on four consecutive days in one year, or the time between two of the earthquakes could be seven months.

This is the Poisson process, and the probability of each event can be calculated.

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It is important that a Poisson process meets the following criteria:

  • The events should be independent of each other. So, the occurrence of one event should not affect the probability of another event occurring.
  • The average rate of the events, i.e., events per time period are constant.
  • Two events should not occur at the same time.

Read: Probability Distribution

Poisson Distribution

Named after the French mathematician Siméon Denis Poisson, Poisson distribution is a discrete probability distribution used to predict the probability of particular events taking place when the average rate of the event is known. In the above example, Poisson distribution can be used to predict the probability of an earthquake occurring at a given time in the year.

It can also be used to predict the event occurrence in various other specified intervals like area, volume, or distance.

The Poisson distribution probability mass function provides the probability of observing k events in a time period when the given length of the period and the average events per time is given. The formula is as follows:

P (k events in interval) = e-λ * λk/k!

Here λ, lambda, is the rate parameter, k is the number of times an event occurs during the time period, e is Euler’s number, and k! is the factorial of k.

Using a simple example, we can see how the probability can be calculated. If the average number of earthquakes striking a city is 2 per year, let us calculate the probability that 3 earthquakes will strike the city in the next year.

Here, k is 3, λ is 2, and e is the Euler’s number, i.e., 2.71828. Plugging these values in the above-given equation, we get P equal to 0.180. This means the probability is 18%. We can conclude that the probability of the city being struck by 3 earthquakes next year is 18%.

Properties Of Poisson Distribution

  • The mean of a Poisson distributed random variable is λ. This is also the expected value.
  • The variance of a Poisson distributed random variable is also the same as the mean, λ.
  • The number of trials in a Poisson distribution can be extremely large. Thus, it can be close to infinity.
  • The constant probability of success in each trial is minimal. Thus, it is close to zero.
  • Since Poisson distribution is characterised by only one parameter λ, it is also known as uni-parametric distribution.
  • Similar to Binomial distribution, Poisson distribution can be unimodal or bi-modal, depending on the rate parameter, λ. If it is a non-integer, then the distribution will be uni-modal, and if it is an integer, then it will be bi-modal.

Poisson Distribution Examples

There are many sectors where Poisson distribution can be used for predicting the probabilities of an event. It is used in many scientific fields and is also popular in the business sector. A few of the examples are stated below.

1. Checking for the amount of a product needed throughout a year. If a business/ supermarket/ store knows the average amount of the products used in a year by their customers, they can use the Poisson distribution model to predict in which month the product sells more. This can help them store the required amount of the product and prevent their losses.

2. Checking for customer service staffing. If the firm can calculate the average number of calls in a day that need more than fifteen minutes to handle, they can use the model to predict the maximum number of calls per hour that require more than fifteen minutes. By calculating this, they can evaluate if they need more staff.

3. It can be used to predict the probability of an occurrence of floods, storms, and other natural disasters. This can be possible if the average number of such disasters per year is known. With these predictions, along with other technological applications, it is possible to avoid human and property losses for many countries or regions.

4. It can also be used in the financial sectors, but these are not necessarily always accurate. This can help in providing an estimate of the probability of how the stock markets will rise or fall at a particular time.

5. The Poisson distribution model can also be used in physics, biology, astronomy, etc. for predicting the probability of meteorites entering the Earth’s atmosphere and being visible in particular regions of the world.

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A popular topic in statistics, Poisson distribution was thoroughly explained through different sections in this article. It is an important topic to understand for students and professionals interested in learning about statistics and probability.

The model can be used in real life and in various subjects like physics, biology, astronomy, business, finance etc., to estimate the probability of an event occurring as mentioned in the examples. Similar topics in statistics, data science, machine learning etc. can be found on upGrad, which will help one expand their learning and apply these concepts to various problems.

If you’re interested to learn more about machine learning, check out IIIT-B & upGrad’s PG Diploma in Machine Learning & AI which is designed for working professionals and offers 450+ hours of rigorous training, 30+ case studies & assignments, IIIT-B Alumni status, 5+ practical hands-on capstone projects & job assistance with top firms.

How is the Poisson process different from the Poisson distribution?

A Poisson process is a model for a series of discrete events in which the average time between occurrences is known but the exact timing is unknown. A Poisson distribution, on the other hand, is a discrete probability distribution that describes the likelihood of events having a Poisson process happening in a given time period. There is an element of occurrences as a sequence in time when discussing the Poisson process, but there is no such element when discussing random variables and their distribution in the Poisson distribution, and we just have a random variable with its associated distribution.

What is meant by a Poisson regression model?

The Poisson regression model is just an example of a generalized linear model. A Poisson regression model is used to model count data and contingency tables. In the case of count models, there are various Poisson regression adjustments that are useful. Given one or more independent factors, Poisson regression is used to predict a dependent variable made up of count data. The variable that we aim to predict is known as the dependent variable.

How is Poisson distribution different from binomial distribution?

Both the distributions come under the umbrella of probability. Binomial distribution refers to the probability of repeating a certain number of trials in a given set of data. The Poisson distribution, on the other hand, explains the distribution of binary data from an infinite sample and specifies the number of independent events that occur at random during a particular time period.

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