The Invention of the Probability Density Function: A Historical Investigation

The concept of the probability density function (PDF), a cornerstone of modern probability theory, has an intriguing history. Early mathematicians approached the topic through discrete distributions and approximation methods, but it wasn't until the contributions of several key figures—Abraham de Moivre, Carl Friedrich Gauss, Pierre-Simon Laplace, and Thomas Bayes—that the PDF began to take shape as we know it today.

The Pioneering Work of Abraham de Moivre

The story begins with Abraham de Moivre, a French mathematician who made significant contributions to probability theory in the 18th century. De Moivre is often credited with the invention of the normal distribution through his discovery of the normal approximation to the binomial distribution. However, de Moivre lacked the concept of a continuous distribution. His approach relied on approximating probabilities using integrals, a method that fell short of fully defining the PDF as we understand it today. If de Moivre had conceived of the integral as representing the density of a continuous distribution, history might have attributed the invention of the normal distribution to him. Unfortunately, this was not the case.

Carl Friedrich Gauss and Robert Adrain: The True Inceptors?

While de Moivre took a crucial step, the accolade for inventing the normal distribution rightly goes to Carl Friedrich Gauss and should be shared with Robert Adrain. Gauss published his famous theorem on the normal distribution in Theoria Combinationis Observationum Erroribus Minimis Obnoxiae in 1809, and Adrain provided an independent proof in 1808. Their work solidified the continuous nature of the distribution, a key advancement in the evolution of the PDF.

Pierre-Simon Laplace: The Advent of Continuous Distributions

The true pioneer in the use of continuous distributions appears to be Pierre-Simon Laplace. In his monumental work, Théorie Analytique des Probabilités, Laplace utilized uniform triangular and exponential distributions. Although Bayes had previously used uniform and beta distributions, Laplace's work on continuous distributions represented a significant advancement in the field. His contributions laid the foundation for understanding and applying continuous probability distributions.

Thomas Bayes: An Earlier Influence

Thomas Bayes, an 18th-century mathematician and philosopher, also deserves recognition for his work with continuous distributions, particularly uniform and beta distributions. Bayes' insights into these distributions predate Laplace's and suggest that the idea of the PDF was floating around in the scientific community during this period.

Further Reading: Historical Insights from Anders Hald

The history of the PDF is rich and complex, with various mathematicians making crucial contributions over time. For a more in-depth exploration, Anders Hald's two seminal works provide valuable insights. His book A History of Mathematical Statistics from 1750 to 1930 offers an extensive account of the development of statistical methods during this period, while A History of Probability and Statistics and Their Applications before 1750 delves into the earlier origins of probability theory and its applications.

These resources contain in-depth bibliographic references and a wealth of historical data, making them invaluable for anyone interested in the evolution of probability and statistics. By examining the contributions of de Moivre, Gauss, Laplace, and Bayes, we can better understand the foundational development of the probability density function.