This should remind you of your probability classes during college years. Can you solve this problem in 30 minutes? This could make for an interesting job interview question.

**Problem**

Points are randomly distributed on the plane, with an average of *m* points per unit area. A circle of radius *R* is drawn around each point. What is the proportion of the plane covered by these (possibly overlapping) circles?

What if circles can have two different radii, either *R* = *r*, or *R* = *s*, with same probability? What if *R* is a random variable, so that we are dealing with random circles? Before reading further, try to solve the problem yourself.

**Solution**

The points are distributed according to a Poisson point process of intensity *m*. The chance that an arbitrary point *x* in the plane is not covered by any circle, is the chance that there is zero point from the process, in a circle of radius *R* centered at *x*. This is equal to exp(-*m *p *R^2*). Thus the proportion of the plane covered by the circles is

If circles have radii equal to *r* or *s*, it is like having two independent (superimposed) Poisson processes, each with intensity *m */ 2, one for each type of circle. The chance that *x* is not covered by any circle is thus a product of two probabilities,

If *R* is a positive random variable and *E* denotes its expectation, then the general solution is

You can easily simulate a large number of these circles over a broad area, and then, pick up 1,000 random points and see how many of them are covered by at least one circle, to check whether your solution is correct or not. You can also use these simulations to get an approximation for exp(-p).

**Application**

The formula for *p*(*m*,*R*) and confidence intervals obtained for its value via simulations under the assumption of pure randomness (Poisson process), can be used to check if a process is really random. For instance, are Moon's craters spread randomly? They might not, if large meteorites break up in small pieces before impact, resulting in clustered craters. In that case, the area covered by the (overlapping) craters might be smaller than theoretically expected.

**DSC Resources**

- Services: Hire a Data Scientist | Search DSC | Classifieds | Find a Job
- Contributors: Post a Blog | Ask a Question
- Follow us: @DataScienceCtrl | @AnalyticBridge

Popular Articles

© 2020 Data Science Central ® Powered by

Badges | Report an Issue | Privacy Policy | Terms of Service

**Most Popular Content on DSC**

To not miss this type of content in the future, subscribe to our newsletter.

- Book: Statistics -- New Foundations, Toolbox, and Machine Learning Recipes
- Book: Classification and Regression In a Weekend - With Python
- Book: Applied Stochastic Processes
- Long-range Correlations in Time Series: Modeling, Testing, Case Study
- How to Automatically Determine the Number of Clusters in your Data
- New Machine Learning Cheat Sheet | Old one
- Confidence Intervals Without Pain - With Resampling
- Advanced Machine Learning with Basic Excel
- New Perspectives on Statistical Distributions and Deep Learning
- Fascinating New Results in the Theory of Randomness
- Fast Combinatorial Feature Selection

**Other popular resources**

- Comprehensive Repository of Data Science and ML Resources
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- 100 Data Science Interview Questions and Answers
- Cheat Sheets | Curated Articles | Search | Jobs | Courses
- Post a Blog | Forum Questions | Books | Salaries | News

**Archives:** 2008-2014 |
2015-2016 |
2017-2019 |
Book 1 |
Book 2 |
More

**Most popular articles**

- Free Book and Resources for DSC Members
- New Perspectives on Statistical Distributions and Deep Learning
- Time series, Growth Modeling and Data Science Wizardy
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- How to Automatically Determine the Number of Clusters in your Data
- Fascinating New Results in the Theory of Randomness
- Hire a Data Scientist | Search DSC | Find a Job
- Post a Blog | Forum Questions

## You need to be a member of Data Science Central to add comments!

Join Data Science Central