You must, my dear Friend, make allowances for the visit I did not pay you: for I set out last month on this wished-for pilgrimage, not with M. Moultou, who is too poor a walker, but with M. de Sauttern; but the deterrent of bad weather which delayed us several days in an inn, my weakness, and the length of the journey, made me give up on it, much as I wished for it, and we retraced our steps after a ten days’ absence which took us no farther than Estavayé. I have not given up hope of being more fortunate another time; but my travel companion has left, and I must admit to you that in my state I lack the courage to tackle alone a trip of forty leagues there, and as many back.

An extensive list of special univariate distributions is studied, and their main properties analyzed. They are the distributions most often encountered in statistics, and we relate them to one another. Some of these distributions arise out of natural phenomena or have attractive special properties which are explored in the exercises. We also study general classifications of distributions and the ensuing properties, including the exponential family, infinitely divisible distributions, and stable distributions. Distributions contain an inherent amount of information and entropy, which we quantify.

A Sketch of the History of Psychiatry in the Nineteenth Century: Introduction to Chapters 9 and 10

As made clear in the American Psychiatric Association's Diagnostic and Statistical Manual of Mental Disorders (DSM–V–TR) (2013, see p. 164), the subject matter of psychiatry is pathological thinking and behavior. Psychology is frequently divided roughly into the study of thinking and the study of behavior, two disciplines that are closely related to each other. As we shall see in Chapters 9 and 10, some interactions have occurred between the two disciplines, but on the whole they have developed independently of each other. I shall therefore not go into a detailed study of the history of psychiatry. To understand the development of clinical psychology, however, and as a background for Chapters 9 and 10, I shall present a sketch of the history of psychiatry in the 1800s.

Mental illness is universal among the peoples of the world, as shown in Kaplan and Sadock's Synopsis of Psychiatry (Sadock and Sadock, 2007). Although the prevalence and manifestations of mental disorders vary with culture and history, mental illness is not a myth created by psychiatrists, or explicable as deviations from social norms or as stigmatization of specific groups of people. Therefore, as Brendan and Winnifred Maher (1985a, b) point out, while the names of mental disorders have changed through Western history, our ancestors were plagued by many of the same types of mental disorders as afflict us today. Thus, when psychiatry was being established as a specialty of modern medicine at the beginning of the 1800s, its pioneers were confronted with many of the same mental disorders as are treated in modern healthcare: intellectual and developmental disabilities, senile dementia, the effects of alcohol abuse, schizophrenia, bipolar disorder, personality disorders, and obsessive-compulsive disorders as well as other anxiety disorders.

The development of psychiatry as a scientific discipline probably owes much to the general progress made in medicine. But as Henri Ellenberger (1970, p. 197) shows, the ideas of the Enlightenment may also have contributed to its advance, by dispelling the medieval view of mental disorders as caused by evil spirits and encouraging a more respectful view of the mentally ill.

If a vector of variates is transformed into anothervector, what is the resulting distribution? This chapter details the main methods of making such transformations and obtaining the new distribution, density, and characteristic functions. Although the proof of the transformation theorem for densities is not usually given in statistics textbooks, we use the shortcut of conditioning to give a statistical proof. Applications of the three methods range from simple transformation (including convolutions) to products and ratios. General transformations are also studied. These include rotations of vectors (and their Jacobian), transformations from uniform variates to others (useful for generating simulated samples in Monte-Carlo studies) via the probability integral transformation (PIT), exponential tilting, and others. Extreme-value distributions are studied, to complement the study of sample averages. The chapter concludes by revisiting the copula, which transforms the marginals into a joint distribution.

This appendix collects mathematical tools that are needed in the main text. In addition, it gives a brief description of some essential background topics. It is assumed that the reader knows elementary calculus. The topics are grouped in four sections. First, we consider some useful methods of indirect proofs. Second, we introduce basic results for complex numbers and polynomials. The third topic concerns series expansions. Finally, some further calculus is presented, including difference calculus, Stieltjes integrals, and multivariable constrained optimization (covering also the case of inequality constraints).

Innovations in Psychology in the United States: Introduction to Chapters 12–15

At the beginning of the twentieth century, its prodigious expansion in agriculture, industry, and commerce had made the United States an economic superpower. Its involvement in World War I was crucial to the war's outcome and also demonstrated the country's military superiority. Heavy investment in the school system and universities led to a blossoming of US science during the first decades of the twentieth century, making the North Americans independent of European culture and science.

Technological development continued its rapid growth, greatly influencing US society. The latter half of the 1800s was a period in which a number of new technological inventions appeared. The United States had already begun contributing at the time of the Civil War, and at the end of the nineteenth century, it had created technological products to revolutionize life at a rate that surprised the rest of the world. Electric lights and the telephone were mass-produced at the end of the 1800s, and the beginning of the 1900s saw the arrival of cars, refrigerators, and radios. Europe and eventually Japan were developing innovations as well, but the Americans were way ahead, as they continued to be throughout the twentieth century. Interest in technology became widespread in the United States and probably contributed to an emphasis on control over understanding in science, and perhaps strengthened the desire to account for psychological phenomena in mechanistic terms.

Empirical psychology was still at an initial stage at the beginning of the 1900s. It had little to offer in the way of new psychological insight, and perhaps more importantly, it had proved to be of little practical use. Feeling more independent of European science and thinking and believing in their own strength, US psychologists began developing their own versions of psychology. In the 1920s, they began laying the foundation for social psychology as a new subarea of psychology. In the next two decades it expanded into a broad field, and I shall treat its development in a separate chapter, Chapter 14.

The clinical and personality psychology of European neurologists was introduced to the United States in the first decades of the 1900s. Inspired by the success of measurements in the study of intelligence, US psychologists in the 1920s began to construct tests for the assessment of personality characteristics.

In the previous chapter, we introduced distributions and density functions as two alternative methods to characterize the randomness of variates. In this chapter, we introduce the final method considered in this book, and relate it to the previous two: the moments of a variate (including mean, variance, and higher-order moments). The condition for the existence of moments is explained and justified mathematically. These moments can be summarized by means of "generating functions". We define moment-generating functions (m.g.f.s). Cumulants and their generating functions are introduced. Characteristic functions (c.f.s) always exist, even if some moments do not, and they identify uniquely the distribution of the variate, so we define c.f.s and the inversion theorem required to transform them into the c.d.f. of a variate. We also study the main inequalities satisfied by moments, such as those resulting from transformations (Jensen) or from comparing moments to probabilities (Markov, Chebyshev). We also show that the mean, median, and mode need not be linked by inequalities, as previously thought.