The Yule-Simon distribution, also known as the Yule distribution, is a highly skewed discrete probability distribution named in honor of George Udny Yule — known for proposing the preferential attachment model for random graphs — and Herbert A. Simon, who received the 1978 Nobel Prize in economics.
The Yule-Simon distribution is a power law distribution used to represent the frequency of specific events within a population. This distribution is commonly used for modeling the distribution of words in a language, the citation frequency of scientific papers, and the distribution of city populations by size.
Yule-Simon distribution properties
Several equivalent forms for the Probability mass function (PMF) exist:
- x is an integer
- Γ is the gamma function
- Β is the beta function
- α can be estimated with a fixed point algorithm .
The Cumulative distribution function (CDF) for the Yule-Simon distribution is P(x) = 1 – xΒ(x, α + 1).
- The Yule distribution is a special case of the beta geometric distribution, when Β = 1 .
- The Waring distribution is a generalization of the Yule distribution.
- For large x-values in the tails, the Zipf distribution and the Yule-Simon distribution are indistinguishable.
History of the Yule-Simon distribution
George Udny Yule was a British statistician known for his significant contributions to the field of statistics and his pioneering work in time series analysis, regression analysis, and the study of probability distributions. Initially, Yule  discussed the distribution, applying it to the distribution of biological genera based on the number of species within each genus. He applied this distribution to analyze the highly skewed pattern observed in the data.
Some of his other notable achievements include:
- Yule Process: Yule proposed an autoregressive model, known as the Yule process or Yule-Walker process, for analyzing time series data. It laid the foundation for modern time series analysis and forecasting methods.
- Yule’s Method of Colligation: Yule developed a technique called “Method of Colligation” to analyze the relationship between two variables. This method is considered an early form of regression analysis.
- Yule’s Paradox: He identified a statistical phenomenon known as Simpson’s Paradox before E.H. Simpson. Yule’s Paradox occurs when an observed relationship between two variables reverses or disappears when a third variable is accounted for in the analysis.
- Association and Contingency: Yule made substantial contributions to the study of association and contingency in cross-classified data. He introduced concepts like the coefficient of association and the coefficient of colligation, which are widely used in statistical analysis today.
Overall, George Udny Yule’s work has had a lasting impact on the field of statistics, with his contributions continuing to influence research and methodology in various disciplines.
Herbert A. Simon (1916-2001) was an American social scientist, economist, cognitive psychologist, and computer scientist, known for his extensive work in the fields of artificial intelligence, economics, decision-making, problem-solving, and organizational theory. He received the Nobel Prize in Economic Sciences in 1978 for his pioneering research on decision-making processes within economic organizations.
Throughout his career, Herbert A. Simon received numerous awards and honors, including the Turing Award (1975) and the National Medal of Science (1986). His interdisciplinary work has had a profound impact on various fields, including economics, psychology, computer science, and management. Simon  independently rediscovered the “Yule” distribution and used it to investigate city populations, income distributions, and word frequency in publications . Although Simon proposed the name Yule Distribution, it is now more frequently referred to as the Yule-Simon distribution . Since its inception, the Yule-Simon distribution has been used in various fields such as linguistics, economics, and bibliometrics, among others, to model the frequency of specific events within a population.
Simon characterized the Yule-Simon distribution as “J-shaped, or at least highly skewed, with very long upper tails” (p. 425), which is indicative of a negative exponential distribution.
Some of Simon’s other notable achievements and contributions include:
- Bounded Rationality: Simon introduced the concept of bounded rationality, which suggests that individuals make decisions based on their cognitive limitations, available information, and time constraints. This idea challenged the traditional economic assumption of perfect rationality and contributed to the development of behavioral economics.
- Satisficing: Simon proposed the term “satisficing” to describe a decision-making strategy where individuals choose the first option that meets their minimum requirements or criteria, rather than seeking the optimal solution. Satisficing is a key component of bounded rationality.
- Artificial Intelligence: Simon was a pioneer in the field of artificial intelligence (AI). Along with Allen Newell, he developed the Logic Theorist, the first AI program capable of proving mathematical theorems. Their work laid the foundation for future AI research, including expert systems and cognitive architectures.
- Organizational Theory and Decision-Making: Simon made significant contributions to organizational theory, particularly in understanding decision-making processes within organizations. His book, “Administrative Behavior” (1947), is considered a seminal work in this area.
The Yule-Simon distribution is used to model various phenomena, including the “superstar effect,” where a select few individuals dominate their respective fields and earn the majority of the income. This phenomenon is also referred to as “cumulative advantage.” For instance, consider a scenario where Brad Pitt and Jane Smith compete for a lead role in a film; the obvious choice would be Brad Pitt due to his fame. Brad Pitt would earn significantly more than Jane Smith, simply because he is well-known. Consequently, Brad Pitt would receive more phone calls, additional offers for paid appearances, and if he were to write an autobiography he would likely secure millions for it. In essence, he has a cumulative advantage over Jane Smith, who would struggle to gain recognition (or compensation), even if her abilities were on par with Brad Pitt’s.
 Yule, G. U. (1925). “A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S”. Philosophical Transactions of the Royal Society B. 213 (402–410): 21–87.
 King, M. (2017). Statistics: A Practical Approach for Process Control Engineers. John Wiley and Sons.
 Garcia Garcia, J. (2011). “A fixed-point algorithm to estimate the Yule-Simon distribution parameter”. Applied Mathematics and Computation. 217 (21): 8560–8566.
 Simon, H. A. (1955). “On a class of skew distribution functions”. Biometrika. 42 (3–4): 425–440.
 Mills, T. (2017). A Statistical Biography of George Udny Yule: A Loafer of the World. Cambridge Scholars Press.
 Hazewinkel, M. (2001). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media.