Group 1
Chen Yanda, Chen Yuxuan, Zhang Rui, Zhou Huinan
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Overview
Poisson Distribution is a discrete probability distribution that is used to model the number of times an event occurs in a fixed interval of time or space. It is named after the French mathematician Siméon Denis Poisson (1781–1840), who introduced the concept in the 19th century.
The Poisson Distribution is often used to model rare events that occur randomly and independently of each other. Examples of such events include the number of car accidents that occur in a city in a day, the number of customers that arrive at a store in an hour, or the number of defects in a batch of products.
The Poisson Distribution is characterized by a single parameter, λ, which represents the mean number of events that occur in the interval of interest. The probability of getting exactly k events in the interval is given by the Poisson probability mass function:
P(X=k) = (λ^k*e^-λ) / k!
where e is the base of the natural logarithm, and k! denotes the factorial of k.
The Poisson Distribution has several important properties, including:
The mean and variance of the distribution are both equal to λ.
The distribution is skewed to the right if λ is small, and becomes more symmetric as λ increases.
The Poisson Distribution can be approximated by the normal distribution when λ is large.
The Poisson Distribution is widely used in many fields, including physics, biology, engineering, and finance. It is an important tool for modeling and analyzing rare events, and can be used to make predictions and estimate probabilities in a variety of contexts.
In this shiny app, we can compute the probability density function, distribution function and inverse function of the Poisson Distribution, and compute various statistics of the Poisson Distribution, such as expected value, variance, standard deviation, etc., based on user-inputted parameters (e.g., the mean, the value of the random variable, etc.).
In addition, the shiny application plots the probability density function and the cumulative distribution function of the Poisson Distribution and provides interactive controls to easily adjust parameters and present results. By using the Poisson Distributions shiny application, users can understand the concepts and applications of Poisson Distributions more intuitively, while also improving the efficiency and accuracy of calculations, helping users to better analyse and solve practical problems.
Video of Poisson Distribution
This Youtube video is about the Poisson Distribution. One can use this video to provide visual aids and examples to help the reader better understand the concepts and applications of the Poisson Distribution. Videos can also show examples of real-world applications of the Poisson Distribution in fields as diverse as finance, engineering, and biology.
The video breaks down complex mathematical formulas and calculations into simpler terms, making it easier for readers to understand and apply the concepts. In addition, the videos can be paused and rewound as needed, allowing everyone to go at their own pace to fully grasp the material. It also helps you deepen your understanding of this important statistical concept.
Instructions
This shiny app is designed to help students better understand the Poisson Distribution. The app consists of four boards, which are:
Lambda for Poisson Distribution: this board allows students to enter the parameter λ of the Poisson Distribution.This parameter determines the shape of the Poisson Distribution, which represents the average number of events per unit of time or per unit of space.
Sample Size: a slider for the number of samples so that the user can increase the number of samples and see the Histogram of Poisson Sampling getting closer to the Standard Poisson Distribution (this is the law of large numbers)
Histogram of the Poisson Sampling: this panel shows the distribution of a sample drawn from the Poisson Distribution. This histogram helps students to better understand the shape of the Poisson Distribution.
Standard Poisson Distribution: generation of Standard Poisson Distribution bar chart based on variations of lambda
With this shiny app, students can better understand the properties of the Poisson Distribution, including:
The Poisson Distribution is a discrete probability distribution that is used to describe the number of times an event occurs over a period of time or in a certain space.
The parameter λ of the Poisson Distribution represents the average number of times an event occurs per unit of time or per unit of space.
Through the Poisson distribution histogram sample plot and standard Poisson Distribution bar chart, students can better observe that the Poisson Distribution histogram sample plot is getting closer to the Standard Poisson Distribution bar chart according to increasing the number of samples in shiny app. It also allows students to understand the nature and application of the Poisson Distribution, thus increasing their interest in learning.
Below is a link to the specifics:
Another shiny app helps students to calculate the probability of Poisson Distribution so that they can test it while solving real-world problems. The app consists of three boards, which are:
Lambda(λ): this board allows the user to enter the parameter λ of the Poisson Distribution.This parameter determines the shape of the Poisson Distribution, which represents the average number of events occurring per unit of time or per unit of space.
k: The board allows the user to enter a value for k, which can be used to calculate the probability that k is greater than or less than or equal to, etc.
Poisson Distribution Histogram: Different values of λ and k are entered by the user to produce the corresponding histogram of the Poisson Distribution. The user can clearly see the changes in the Poisson Distribution Histogram as λ and k are varied.
By entering different values of λ and k, the user can obtain histograms of the corresponding probabilities and Poisson Distributions. When using this app, the user needs to choose the appropriate probability according to the specific situation.
The following is the link to the Shiny app. Use it complete Tasks below:
Tasks
Part A — Properties of Poisson Distribution：
Q1. Which of the following events is most likely to be modeled by a Poisson Distribution？
A) Number of goals scored in a soccer match
B) Height of individuals in a population
C) Temperature readings throughout a day
D) Number of phone calls received in a call center within a fixed time period
Answer
Answer
The correct answer should be D)
The number of phone calls received in a call centre within a fixed time period is most likely to be modelled with a Poisson distribution. We can know from the definition of the Poisson distribution that the Poisson distribution is typically used to model the number of occurrences of a rare event over a fixed time or spatial interval. In this case, the events are independent of each other and have a constant average occurrence. In the case of a call centre, the number of calls received in a fixed time period is a rare event and its incidence can be assumed to be constant over the time period. Therefore, the number of calls received by a call centre in a fixed time period is well suited to be modelled using the Poisson distribution.
Part B — Applications of Poisson Distribution：
hint：for this part, you can use shiny to enter the parameter including λ and k to get the corresponding graph and checking your answer~
Q1. A new shopping mall is considering setting up an information desk manned by one employee. Based upon information obtained from similar information desks, it is believed that people will arrive at the desk on average at a rate of 15 per hour. It is assumed that the arrivals follow a Poisson Distribution.
1）Calculate the probability that in a given hour, there will be less than 12 people arriving？
2）Calculate the probability that in 30 minutes, there will be more than 6 people arriving？
3）The employee wants to have a 10 minute break but knows that is any one arrives and finds the desk unoccupied there will be a complaint. If the employee does take a 10 minute break, what is the probability that the shopping mall will receive a complaint?
Answer
We determine the values of λ and k by using the formulae in the definition of the Poisson distribution thus substituting it into Eq. Thus getting the answer we want.
For the first question we can check our answer using the probability of Poisson Distribution's shiny app. From the question we know that λ=15,k=12. On the right hand side we can see P(X<12)=0.184751799023931. This is the answer we get.
The second and third questions can be carried out in this same way by entering the values of λ and k, thus generating the probabilities via the shiny app.
Q2. The average number of persons entering hospital casualty per hour is five. Assume the distribution of hospital casualty entries per hour can be approximated by the Poisson Probability Distribution.
1)What is the probability that 7 people will enter hospital casualty in the next hour?
2) What is the probability more than 4 people will enter hospital casualty in the next half an hour?
Answer
The average number of admissions to the hospital per hour is 5, which is the mean and variance of the Poisson Distribution. Using the Poisson probability mass function, the probability that seven people will enter the hospital in the next hour can be calculated by substituting λ and k into the formula.
This is done in the shiny app by entering the values of λ and k, which results in a probability and an image. So that students can check their answers.
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