Elementary Tools for Stochasitc Analysis

Abstract

Lambert Adolf Jacob Quételet (1796–1874), the astronomer, first attempted to ground statistical sociology with the ‘average man’ (l’ homme moyen). He then exhibits an interesting question about the difference between criminals and ordinary people. Even if the probability distribution of crime occurrence is Gaussian, different theoretical backgrounds arise depending on whether crime is regarded as acquired or innate. If

it is considered acquired, there will be a uniform population with a certain probability of becoming a criminal, whereas if all criminals are congenital, there is a probability of being born a criminal. Quételet is also currently well known by the body mass index (BMI) scale, originally called the Quetelet Index. The obesity index is also attributed to him. He proposed that society be modeled using mathematical probability and social statistics, which brought the ‘enthusiasm for normal distributions’ of natural and social scientists in the 19th century. Indeed, from the discoveries of Bernoulli and de Moivre, it came to be thought that large amounts of data on completely chance events could be used to argue for nonchance regularities. Moreover, the large amounts of data that came to be accumulated with the industrial revolution were also thought to apply to population, industry and society. This was the birth moment of social statistics. It came to be thought that distribution laws such as the Gaussian normal curve could be recognised everywhere. In other words, if we can consider ‘normal size’, we can also talk about ‘normal distribution’. Clearly, everyone has the freedom to be or not to be a criminal. However, the probability may be already determined at birth. In this circumstance, unlimited freedom and responsibility could not be guranteed. From the dawn of the Enlightenment, the ideals of freedom and probabilistic design have not always been compatible. The enthusiasm of probability distribution, on the contrary, was rather irrelevant to ethics. As time evolved since the enthusiasim due to Quételet, various problems were addressed, such as ‘the existence of dependence or interaction between individual data, or the occurrence of unusually large scatter and fluctuations’, and the assumption of normality in asset price fluctuations, for example in economic examples, was dismissed. This is nothing but an achievement of econophysics, but this is a very recent one. Finally, we learn the Polya urn process as a good reference mechanism to grasp the evolution of the system.