What is the law of Pearson distribution? The answer to this broad question cannot be simple and concise. The Pearson system was originally designed to model visible distorted observations. At that time, it was well known how to set up a theoretical model to fit the first two cumulants or moments of the observed data: any probability distribution can be directly expanded to form a group of location scales.
Pearson hypothesis on the normal distribution of criteria
With the exception of pathological cases, the location scale can be made so that it corresponds to the observed average value (first cumulant) and variance (second cumulant) in an arbitrary way. However, it was not known how to construct probability distributions in which the asymmetry (standardized third cumulant) and excess (standardized fourth cumulant) could be regulated equally freely. This need became apparent when trying to fit well-known theoretical models to observable data that showed asymmetry.
In the video below, you can familiarize yourself with Pearson's chi-distribution analysis.
History
In his original work, Pearson identified four types of distributions (numbered I to IV) in addition to the normal distribution (which was originally known as type V). The classification depends on whether the distributions are supported on a limited interval, on the semi-axis, or on the entire real line, and whether they were potentially skewed or necessarily symmetrical.
In the second article, two omissions were corrected: he redefined the distribution of type V (initially it was only a normal distribution, but now with an inverse gamut) and introduced a distribution of type VI. Together, the first two articles cover the five main types of the Pearson system (I, III, IV, V, and VI). In a third article, Pearson (1916) introduced additional subtypes.
Concept improvement
Rind invented a simple way to visualize the Pierson system parameter space (or distribution of criteria), which he subsequently adopted. Today, many mathematicians and extras use this method. The types of Pearson distributions are characterized by two quantities, commonly called β1 and β2. The first is the square of asymmetry. The second is the traditional excess, or the fourth standardized moment: β2 = γ2 + 3.
Modern mathematical methods determine the excess γ2 in the form of cumulants instead of moments, so for a normal distribution we have γ2 = 0 and β2 = 3. Here we should follow the historical precedent and use β2. The diagram on the right shows what type the specific Pearson distribution (indicated by the point (β1, β2) belongs to.
Many of the distorted and / or non-mesocurtic distributions we are familiar with today were not yet known in the early 1890s. What is now known as the beta distribution was used by Thomas Bayes as the posterior parameter of Bernoulli's distribution in his 1763 work on the inverse probability.
The beta distribution gained fame due to its presence in the Pearson system and was known until the 1940s as the Pearson Type I distribution. A type II distribution is a special case of type I, but usually it is no longer isolated.
The gamma distribution arose from his work and was known as the normal distribution of Pearson type III, before it acquired its modern name in the 1930s and 1940s. In a scientist's article for 1895, a type IV distribution was presented, which contains the t-distribution of Student, as a special case preceding the subsequent use of William Seeley Gosset for several years. In his 1901 article, a distribution with inverse gamma (type V) and beta primes (type VI) was presented.
Another opinion
According to Horde, Pearson developed the basic form of equation (1) based on the formula for the derivative of the logarithm of the normal distribution density function (which gives a linear division by a quadratic structure). Many experts are still testing the hypothesis on the distribution of Pearson's criteria. And she confirms her effectiveness.
Who was Karl Pearson
Karl Pearson was an English mathematician and biostatist. He is credited with creating a discipline of mathematical statistics. In 1911, he founded the world's first Department of Statistics at University College London and made a significant contribution to biometrics and meteorology. Pearson was also a supporter of social Darwinism and eugenics. He was a protege and biographer of Sir Francis Galton.
Biometrics
Carl Pearson played an important role in creating the school of biometrics, which was a competing theory for describing the evolution and inheritance of the population at the turn of the 20th century. His series of eighteen works, “The Mathematical Contribution to the Theory of Evolution,” established him as the founder of the biometric school of inheritance. In fact, Pearson devoted a lot of time during 1893-1904. the development of statistical methods for biometrics. These methods, which are widely used today for statistical analysis, include chi-square test, standard deviation, correlation and regression coefficients.
The question of heredity
Pearson's law of heredity stated that germplasm consists of elements inherited from parents, as well as from more distant ancestors, the proportion of which varied according to different characteristics. Carl Pearson was a follower of Galton, and although their work differed in some respects, Pearson used a significant number of statistical concepts from his teacher in the formulation of a biometric school for inheritance, such as the law of regression.
School Features
The biometric school, in contrast to Mendeleev’s, was focused not on providing a mechanism of inheritance, but on providing a mathematical description that was not causal in nature. While Galton proposed a discontinuous theory of evolution in which species would have to change through large leaps, rather than small changes that accumulated over time, Pearson pointed out the flaws in this argument and actually used his ideas to develop a continuous theory of evolution. Mendelevites preferred the discontinuous theory of evolution.
While Galton focused mainly on the application of statistical methods to the study of heredity, Pearson and his colleague Weldon expanded their reasoning in this area, variation, correlation of natural and sexual selection.
A look at evolution
For Pearson, the theory of evolution was not intended to identify the biological mechanism that explains the laws of inheritance, while the Mendelian approach declared the gene to be the mechanism of inheritance.
Pearson criticized Bateson and other biologists for not adopting biometric methods in his study of evolution. He condemned scientists who did not focus on the statistical validity of their theories, stating:
“Before we can accept [any cause of progressive change] as a factor, we must not only show its credibility, but, if possible, demonstrate its quantitative ability.”
Biologists have succumbed to “almost metaphysical assumptions about the causes of heredity,” which replaced the process of collecting experimental data, which in fact can allow scientists to narrow down potential theories.
Nature laws
For Pearson, the laws of nature were useful for accurate predictions and for a brief description of trends in observed data. The reason was the experience, "that a certain sequence has occurred and repeated in the past."
Thus, identifying the specific mechanism of genetics was not a worthy occupation for biologists, who should instead focus on mathematical descriptions of empirical data. This partly led to a fierce debate between biometrists and Mendelians, including Bateson.
After the latter rejected one of Pearson's manuscripts describing a new theory of offspring variability or homotype, Pierson and Weldon founded Biometrika in 1902. Although the biometric approach to inheritance ultimately lost Mendel’s view, the methods they developed at that time were vital to the study of biology and evolution today.