Probability Theory #4 — Cauchy distribution

In the last three discussions on probability theory, we looked at discrete distributions. They are characterized by values of outcome that are integers. For example, a yes is 1 and a no is 0. In the case of flipping a coin, a yes is assigned to the head and a no is assigned to the tail when the desired outcome is the head. We do not expect any decimals.

In this discussion, we will look at continuous distributions. Unlike discrete distributions, continuous distributions can take on decimals. For example, temperature can be measured at 0.000000001 intervals and the decimals can be as long as the instrument allows.

So what is Cauchy distribution? I’ll let Bard do the explanation.

Imagine a group of people who are all standing in a line. The Cauchy distribution can be thought of as a model for the distance between each person in the line. Some people will be very close together, while others will be far apart. The scale parameter of the distribution controls how far apart the people are on average. The location parameter of the distribution controls where the center of the line is.

If the scale parameter is large, then the people in the line will be very spread out. This means that there is a high probability of observing extreme values, such as a person who is very far away from everyone else. If the scale parameter is small, then the people in the line will be more closely grouped together. This means that there is a lower probability of observing extreme values.

The location parameter of the distribution controls where the center of the line is. If the location parameter is equal to zero, then the center of the line is at the origin. If the location parameter is not equal to zero, then the center of the line is shifted to the right or left by the amount of the location parameter.

As you may have read in the explanation, Cauchy distribution is very much about recognizing the important role of extreme values in reality. Unlike normal distribution whereby the values are distributed in a manner that is well spread given its variance, Cauchy distribution recognizes the existence of over-dispersion — a term we use to describe the extreme values in coexistence with the more commonly observed values.

Another example of Cauchy distribution is the magnitude of earthquakes. In Japan, minor earthquakes are common across the year but everyone in Japan knows a very big earthquake will come at some point in time. It is not a surprise; we are expecting it to happen but we just do not know when it will happen exactly. In this example, we can correctly assume that the big earthquake is an expected extreme value and all the small earthquakes are the commonly expected values.

Daniel started off his career as a senior list researcher with a British publishing firm. Back then, his role involved contact sourcing through the internet and performed data entry into the Microsoft Dynamic CRM system. (Microsoft Dynamic CRM 3.0) Progressively, he explored the option of using Visual Basic scripting within excel to automate the contact sourcing process.

He successfully developed and implemented the scripts, leading to 95% increase in data entry efficiency. He then moved on to take on the role of a CRM executive with Fuji Xerox Singapore.

As a CRM executive, he liaised with third party vendor for technical enhancement of the CRM system (Microsoft Dynamic CRM 4.0 and 365). He also performs functional enhancement of the CRM system for hundreds of end users.

His notable achievement was the development of the CRM boy that led to 98% improvement in data quality and data integrity in the CRM system. Following his Masters studies in Consumer Insight with Nanyang Business School, he took on the role of an Analytics instructor with Singapore Management University. He prepared class notes and technical walkthrough, and taught Analytics to the undergraduate students from various disciplines. Subsequently, he took on various roles as consultants in the consultancy, manufacturing and information technology industries in Singapore.

He travelled to Paris, London, Sri Lanka, Japan and Malaysia to fulfill his role as a consultant. The cultural and professional exchanges between local and overseas data analytics had given him a very good overview of the expectations and motivations from people around the world. He also had a chance to relocate to the United States for one year, particularly focusing on Operations Management.

Prior to his current freelance status, he took on the role of the Data Science Lead in a Singaporean software company. His primary role was to develop Artificial Intelligence using logic, data science and machine learning techniques through in-depth, full-stacked scripting. He also developed customized Reporting for his customers. In his point of view, 95% of today’s reporting can be automated, which can free up staff from daily manual work.

He holds a Bachelor of Science in Marketing (BSc. Marketing Pass with Merit) from Singapore University of Social Sciences (in which he graduated as a Valedictorian), a Master of Science in Marketing and Consumer Insights (MSc. Marketing and Consumer Insights) from Nanyang Technological University, a Doctor of Business Administration (DBA) from Swiss School of Business and Management.