A critical note on moments of Cauchy distribution based on LOTUS theorem
*G. S. David Sam JayakumarCorresponding authorsamjaya77@gmail.comDepartment of Management StudiesJamal Institute of ManagementJamal Mohamed College (Affiliated to Bharathidasan University)Trichy, Tamil Nadu, 620020, India0000-0002-6630-8783View full profile → , W. Samuelwsm@jmc.eduwsamuel365@gmail.comDepartment of Business AdministrationJamal Mohamed College (Affiliated to Bharathidasan University)Trichy, Tamil Nadu, 620020, India0000-0002-9112-9744View full profile → , Bejoy John Thomasbejoy@rajagiri.eduDepartment of Business AdministrationRajagiri College of Social SciencesRajagiri Business SchoolCochin, Kerala, 683104, India0000-0002-8395-3063View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Aug 2025
- Published Online:
- 08 Sep 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JSMS-1606
- Pages:
- 1–18
Abstract
The moments of Cauchy distribution in statistics are still unexplored and the statisticians already concluded that they are undefined when it involves direct integration. The authors explored the higher order moments of the Cauchy distribution based on LOTUS theorem (Law of the unconscious statistician). At first, the theorem is applied to compute the moments of the normal distribution and the result of the moments are similar when we compared it with the existing results. Hence, this motivates the authors to derive the finite sample moments of Cauchy distribution are numerically displayed. The results derived by the authors are surprising because the real valued random variable of a Cauchy distribution is having finite moments in the complex plane and the moments are expressed in complex numbers. The first ten sample finite moments of the Cauchy distribution are visualized with the help of polar plots to know the magnitude and proximity among the moments.
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References
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