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# Poisson Distribution

Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837.The Poisson distribution is used to model the number of events occurring within a given time interval. The Poisson distribution arises when you count a number of events across time or over an area i.e. it is a count distribution.

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Johanan Collins on 28th Sep 2021.

Applause for all the respondents - Saravanan MR, Mokshesh, Varuna Kakathkar, Amit Kumar, Vijay Krishnan, Johanan Collins, Mohammad Mahmudul Hassan.

## Question

Q 404. What is Poisson Distribution and how can it help us analyze the data? Quote some examples from your work area where data follows this distribution.

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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The Poisson Distribution is named after the French mathematician Simeon Denis Poisson. It is a discrete distribution that gives the probability of the number of events in a fixed interval of space, time, distance, volume, area, etc. These events should take place independent of each other and at a constant rate. The Poisson distribution is generally used where the number of events is very large and the occurrence of these events is rare. In simple terms it can be described as the average rate at which a specific event happens in a specific time frame, the event will follow a Poisson distribution.

Assumptions

k is the number of times the event can occur (k= 0,1,2,3,…..)

The events are independent of each other

The average rate at which events occur, are independent of the occurrences and may vary with time.

Two events cannot occur at the same time.

Examples

The number of calls received in a Call Centre every hour.

The number of decay events that occur from a radioactive source every day.

No of Accidents in a day on Highway No 4.

The number of meteorites greater than 0.5-meter dia that strike Earth in a year.

The number of customers arriving at a Mall between 8 and 9 am.

Number of Tsunamis that hit the East Coast of India in a decade

Number Goals in a match.

Number of Injuries in a football match.

Number of Hat tricks in a cricket match

Examples that are not Poisson Distribution

The arrival of students for a class

The arrival of spectators for a match.

These are not Poisson Distributions since the rate of arrival is not constant and the events are not independent of each other since students and spectators come in batches.

Occurrences of earthquakes may not follow a Poisson distribution since the aftershocks may not be independent of each other.

Formula

The formula for Poisson distribution is

Where

k is the number of occurrences,

e = 2.71828 (Euler’s Number)

Lamda = E(X) [The mean] = Variance(X)

Lamda can also be taken as the rate for the number of events.

Poisson Distribution in Management

Poisson Distribution is generally used to improve operational efficiency.

Poisson Distribution is used in Queueing Theory as the arrival rate, whereas the Exponential Distribution is used in the Service times.

Managing Demand. Study of the demand through the Poisson Distribution can help in developing complementary services, managing the reservation system and overbookings such as in hotels and airlines, segmenting demand, offering price incentives, or promoting off-peak demand.

Managing Supply. Similarly, the study of the supply can help in sharing capacity, take decisions on cross-training of staff, using part-time employees, increasing customer participation, scheduling work shifts, or creating adjustable capacity.

References

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Poisson Distribution:

The French Mathematician Siméon Denis Poisson who provided a systematic mathematical way to solve such a problem in the 18th century about a large number of possible events where the occurrence is very rare or small. It shall be used for discrete distributions which measure the probability of possible events in a particular time period. In other words, it describes the random occurrence of events in a particular time period.

The Poisson distribution is also called the law of small numbers. The reason behind that is the occurrence of the event is very rare in the large number of events.

The number of occurrences fluctuate about its mean  λ and with the standard deviation of the square root of  λ . The correlation between mean and standard deviation counting independently is also useful.

The confidence interval of Poisson mean calculated by using relationship between Poisson and Chi-square distribution,

In the Poisson process, The data are discrete, independent to each other and mutually exclusive.

The Poisson and binomial distribution are applicable for attribute and discrete data.

The Poisson is an approximation of binomial distribution. The binomial is used to calculate the probability of the number of successes obtained from n trial with p probability, whereas in Poisson distribution is a special case of binomial distribution as n will be infinity and while keeping the expected number of successes remain the same.

Characteristics:

• This distribution is generally used for describing the probability distribution of an event with respect to time or space

• Suitable for analyzing situations where the no. of trials (remember sample size in Binomial distribution) is very large (tending towards infinity) and probability of success in each trial is very small (tending towards zero).

• Hence applicable for predicting occurrence of relatively rare events like plane crashes, car accidents etc. and therefore used in Insurance industry

• Can be used for prediction of no of defects

Poisson distribution formula is as below:

Where:

λ(lambda)= Mean number of occurrences during interval

x= number of occurrences desire

e= The base of the natural log (equals to 2.71828)

Mean of a Poisson distribution μ= λ

Std. dev of a Poisson distribution σ= √λ

The width of the distribution increases with μ, as it indicates that the uncertainty will increase with an increase of x.

Conditions for Poisson distribution:

• Events are independent of each other.

• Rate of event is constant (events per period, area, volume, length etc)

• Two events cannot occur parallel or simultaneous

As an example 1:  An automobile company has average defects per unit (DPU) 0f 0.3, what is the probability they would find 3 defects in a vehicle? Here we can use Poisson distribution to calculate the probability.

We can do this Manually, or some online calculator or standard table as well. Please refer to the below methods for more understanding.

1. Manually calculated:

1. The same example did in the Poisson calculator as below:

1. The same can be obtained from Poisson distribution table:

As an example 2: There are an average of 0.3 accidents per day on the highway between 2 cities, what is the probability that there will be 3 or more accidents in a day? Here we can use Poisson distribution to calculate the probability.

This problem also can be calculated based on the above methods.

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What is Poisson Distribution?

Poisson distributions is amongst one of the most practical distributions in answering many of the questions of the modern world. Being used more than a century now, this probability distribution has been useful in solving various problems from medical, banking, agriculture, defence, mining, space research, service and manufacturing industries.

Poisson distribution is named in honour of French mathematician and physicist Simeon Denis Poisson. It is a discrete distribution because it shows probabilities of countable distinct values. In simpler terms, this distribution does not take all values in a continuous range. The distribution takes the values 0, 1, 2, 3, 4, etc. with no decimals and fractions.

The formula of Poisson distribution is as below:

Where,

x = 0, 1, 2, 3….

λ is a real number and the expected value of x

e ≈ 2.718

Understanding Poisson distribution

A Poisson distribution is useful in estimating the probability that something will happen "X" number of times within a given time period. For example, if the average number of students who bunk the tuition class on a public holiday is 5, a Poisson distribution can answer questions such as, "What is the probability that more than 10 students will bunk the classes on a given public holiday?" The application of the Poisson distribution thereby enables tuition teachers to introduce optimal lecture schedules that would not work with, say, a normal distribution.

Before applying the Poisson distribution, there are few conditions to be satisfied as below:

1. x is the number of event that occurs in an interval and x can take values 0, 1, 2, ....

2. The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.

3. The probability an event occurs is the same throughout the entire time interval.

Understanding Poisson distribution and its application from work area (Insurance)

(Numbers used are dummy for educational purpose)

The number of death claims received per day at an insurance company follows a Poisson distribution. The company receives an average of 3 death claims per day. The claims department needs an expert claims assessor to assess each case and settle the claims within promised timelines. The current staff size is capable to manage daily 5 cases. The department manager is allowed to recruit more staff if the probability of receiving more than maximum manageable cases exceeds by 20%. As a department manager, you would want to evaluate if you are eligible for fresh recruitment in department or not. You may use Poisson probability distribution to answer this question.

In this case, λ =3, hence the Poisson distribution for our case becomes as below:

----- (equation 1)

The question states you to calculate the probability of receiving more than maximum manageable cases i.e. more than or equal to 6 cases (X≥6).

In this case, the probability using Poisson distribution will be calculated as below:

Inserting x = 0,1,2,3,4,5 in equation 1 above, we get answer as below.

Since the probability of receiving more than manageable cases is less than 20%, the department manager is not eligible for a new recruitment in the department.

Visually, one can represent the above Poisson distribution problem as below:

Bibliography:

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Poisson Distribution

The Poisson Distribution is a discrete function used in probability theory statistics to show how many times an event is likely to occur over a specific period. This describes an average probability of an event happening per unit (Per unit of time, cycle, event). For example, Event “A” happens, on an average “x” times per hour. It was originated by a French Mathematician Simon Denis Poisson in 1837. This distribution is used for rare events. Poisson distribution is very similar to Binomial Distribution. Binomial describes the exact probability happening number of times out of x (i.e. 10 times out of 100 or 90 times out of 1000).

Fig: Poisson Distributions, (λ) Also known as μ is expected number of events occurrences on the horizontal axis.

Poisson Distribution can be used as:

1.       Determining variations measure will likely be from that average number of occurrences.

2.       To determine the probability of maximum and minimum number of times the event will occur within the specific time period.

Conditions of Poisson Distribution:

1.       Number of possible occurrences is not limited.

2.       Occurrences are independent.

Average number of occurrences is the same from unit to unit

Fig: Computing the Poisson Distribution with parameter lambda (λ) or μ = 4

Fig:  The Normal distribution closely approximates with the Poisson distribution

Mean and Variances of Poisson Distribution:

When λ is large the Poisson Distribution can be approximately by the normal distribution with μ = λ and σ2 = λ (where, σ is Standard deviation and μ is Mean)

μ is the average number of occurrence in a given time interval or region in the Poisson distribution, then the mean and variance of the Poisson distribution are both equal to μ.

Note:  In a Poisson distribution, μ is needed to determine the probability of an event.

Poisson Characteristics:

Poisson Distribution focuses on the number of discrete events or occurrences over a specified interval like Time, Length, Distance etc.

1.       Discrete outcomes like (x = 0, 1, 2, 3….)

2.       The number of occurrences in each interval ranging from zero to infinity (Theoretically)

3.       Describes the distribution of infrequent events or rare events

4.       Each events in independent of the other events

5.       Describe discrete events over an interval (Time, Distance etc.)

6.       Expected number of occurrences are assumed to be constant throughout the experiment.

Given above condition, x is a random variable and distribution of x is a Poisson Distribution.

Birth defects and Genetic mutations.   Traffic Flow and ideal gap distance.

Failure of a machine in one month.       Queuing theory (Waiting time problem).

The demand of a product in equals intervals of time. Arrival pattern in a departmental store.

The occurrence of defects in a manufacturing units.  Car Accidents.

Examples of Poisson Distribution:

1. Check for adequate customer service staffing:

Average number of services calls per hour requires 10 minutes to handle. By applying Poisson distribution formula, we can find the probability of maximum number of calls per hour that might come more than 10 minutes per hours. Then, we will evaluate the staffing are adequate to handle the calls of 10+ minute without making customer wait on hold.

2. Evaluate to financial viability to keep a store 24 hours a day:

Calculating the average number of sales made by store from mid night to 8:00 am for a night shift of company ElectroCom. Using the Poisson distribution formula, we can calculate the lowest selling from store during that time, then we can determine the total running cost of wages, electricity to decide the store needs to opens 24 hours a day to make financial viable or not for the company.

√ No of accidents √ No of deaths in floods √No of mistakes per page committed by typist

√ No of goals games of football and Hockey √ No of arrivals at a Car wash in one hour

√ No of network failures per day √ Number of hungry persons entering McDonald restaurant per day.

√ No of births, deaths, marriages, divorces, suicides over a given period of time.

Description of a Problem:

At ElectroCom Ideas & Technologies Ltd, an average daily sales volume of Diesel Generator in whole the country is Five (5). Calculate the probability of ElectroCom Ideas & Technologies Ltd that exactly nine (9) diesel generators will be sold tomorrow.

Solution:

Poisson distribution formula in below equation where the mean (Average) number of events within a specified time period is designated by μ. The probability formula is

P (x; μ) = (e) (μx) / x!

μ=5, Since five (5) diesel generator is the daily average sales by all the distributors of ElectroCom.

x=9, Because we want to solve for the probability of Nine (9) diesel generators being sold tomorrow.

e= 2.72 1828 since e is a constant equal to approximately 2.71828

Putting these values in the below formula:

P (x; μ) = (e) (μx) / x!

= (2.71828-5) (59) / 9!

= (0.0067) (1953125) / (3262880) = 0.036

Thus, 3.6% is the probability of selling nine (9) no’s diesel generators being sold tomorrow.

At the end, a wide range of industries including medicine, astronomy, business, and sports etc. can use the Poisson distribution to examine the capability to take necessary steps to improve their operational efficiency.  By analyzing Poisson distribution, the management able to better handle the peak periods by necessary staffing rearrangement and many other operational activities. Also can get into more insights to improve their operational efficiency.

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Before explaining about the Poisson's distribution, let me brief about data types.

For any process before making any analysis, first we need to identify the type of data. Data can be divided as Continuous and Discrete. The Discrete data is further categorized into Binomial and Poisson distribution. Unlike the Binomial distribution, which has only 2 outcomes (Success or fail, Good or Bad, Ok or Not Ok etc.),Poisson distribution focusses on number of discrete occurrences over a pre defined time interval.

Some Facts/Assumptions/Properties of Poisson's distribution:

• The average number of instances those are occurring in a specific area are known
• Outcomes are random.
• The outcomes of interest are rarely related to the possible outcome.
• Poisson's Formula for finding out the Probability:

Lambda=Population Mean=n*p (Sample Size * Probability)

Examples:

1. The number of component failures for every "x" hours
2. The number of defective items for every 1000 samples
3. The number of shipments booked in a website in a day
4. The number of transaction per day in a bank etc.

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In Statistics, Poisson distribution was named after French mathematician Siméon Denis Poisson.
Poisson distribution is a tool which helps for predicting the probability of certain events happening when we know how often particular event had occurred in the past already.

It has condition as Events occur independently. One Example of the Poisson distribution usage is calls coming in call center & capacity planning is being done based on the Poisson distribution of the incoming calls. The mean & variations of the Poisson distribution is same.

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Poisson distribution , named after  French mathematician Siméon Denis Poisson is a probability distribution used to show number of times an event occurred in a given time period. The poissons function is a discrete variable in that it can take only whole number values and cannot take fractions or decimals. This distribution can be used to estimate how likely it is that something will happen "X" number of times.

• e is Euler's number (e = 2.7182..)

• x is the number of occurrences

• λ is equal to the expected value of x when that is equal to its variance

Some examples at my workplace

·         Number of Luminaires arriving at the packing line between 5 and 6 PM

·         Unplanned absentees from production on any given working day at the factory

·         Orders waiting for dispatch after cut-off time of 7PM

·         Complaint calls received by complaint team on a given day

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Excellent responses to a relatively difficult question (after all it involves data distributions )

Johanan Collins has provided the best answer to this question. However, there are other brilliant answers that are must read - Saravanan MR, Mokshesh and Mohammad Mahmudul Hassan.

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