This chapter outlines the growth process as well as determinants of growth for the cultural sector. We touch upon supply and demand factors that have stimulated growth. To illustrate the growth process across countries, we present data on attendance at the live performing arts in seven European countries and in the United States, Canada, and Australia. Finally, the chapter covers important institutional changes that occurred in the live performing arts sector from the twentieth century and beyond.

An important and ubiquitous continuous distribution is the Normal distribution (also called the Gaussian). Normal distributions occur frequently in statistics, economics, natural sciences, and social sciences. For example, IQs approximately follow a Normal distribution. Men’s heights and weights are approximately Normally distributed, as are women’s heights and weights.

We have alluded to the fact that probability is useful in the performance analysis and design of computer systems. Queueing theory is an area of applied probability which directly targets systems performance. Here the “system” might refer to a computer system, a call center, a healthcare system, a manufacturing system, a banking system, or one of many other examples. Markov chains (particularly continuous-time chains) are just one of many tools used in queueing theory. In this final part of the book, we provide a very brief introduction to queueing theory. For a much more in-depth coverage, see [35].

At this point, we have discussed many discrete and continuous distributions. This chapter shows how we can generate instances of these distributions and others. This is helpful when performing simulations of computer systems, as in Chapter 14. For example, we might have a computer system where the interarrival times of jobs are well modeled by an Exponential distribution and the job sizes (service requirements) are well modeled by a Pareto distribution. To simulate the system, we need to be able to generate instances of Exponential and Pareto random variables.

So far we have only talked about finite-state discrete-time Markov chains (DTMCs) with $M$ states. Now we move on to infinite-state DTMCs. For a Markov chain with an infinite number of states, one can still imagine a transition probability matrix, $\mathbf{P}$, but the matrix has infinite dimension.

In this part of the book, we repeat the material in Part II, but this time we focus on continuous random variables, which can take on an uncountable number of values. Continuous random variables are very relevant to computer systems – how else can we model response time, for example? Working in continuous time also allows us to leverage everything we know about calculus.

Richly illustrated in full colour and packed with examples from every major continent and wetland type, this third edition has been completely rewritten to provide undergraduates with a thoroughly accessible introduction to the basic principles. It divides the world’s wetlands into six principal types and presents six major causal environmental factors, arranged by importance and illustrated with clear examples, making it easy for instructors to plan tailored lectures and field trips and avoid overwhelming students with unnecessary detail. It retains its rigour for more advanced students, with sections on research methods and experiments, and over a thousand classic and contemporary references. Each chapter ends with questions that review the content covered and encourage further investigation. With expanded sections on topical issues such as sea level rise, eutrophication, facilitation and the latest approaches to restoration and conservation, the new edition of this prize-winning textbook is a vital resource for wetland ecology courses.

Richly illustrated in full colour and packed with examples from every major continent and wetland type, this third edition has been completely rewritten to provide undergraduates with a thoroughly accessible introduction to the basic principles. It divides the world’s wetlands into six principal types and presents six major causal environmental factors, arranged by importance and illustrated with clear examples, making it easy for instructors to plan tailored lectures and field trips and avoid overwhelming students with unnecessary detail. It retains its rigour for more advanced students, with sections on research methods and experiments, and over a thousand classic and contemporary references. Each chapter ends with questions that review the content covered and encourage further investigation. With expanded sections on topical issues such as sea level rise, eutrophication, facilitation and the latest approaches to restoration and conservation, the new edition of this prize-winning textbook is a vital resource for wetland ecology courses.

Richly illustrated in full colour and packed with examples from every major continent and wetland type, this third edition has been completely rewritten to provide undergraduates with a thoroughly accessible introduction to the basic principles. It divides the world’s wetlands into six principal types and presents six major causal environmental factors, arranged by importance and illustrated with clear examples, making it easy for instructors to plan tailored lectures and field trips and avoid overwhelming students with unnecessary detail. It retains its rigour for more advanced students, with sections on research methods and experiments, and over a thousand classic and contemporary references. Each chapter ends with questions that review the content covered and encourage further investigation. With expanded sections on topical issues such as sea level rise, eutrophication, facilitation and the latest approaches to restoration and conservation, the new edition of this prize-winning textbook is a vital resource for wetland ecology courses.

This chapter delves deeper into the process by which artists make their creations available to purchasers. We will also focus on why these purchasers decide to acquire art. Ultimately, these decisions constitute the supply of and demand for art, which eventually helps explain both the levels of art prices and the changes in those levels. We describe the market for paintings that consists of both primary and secondary markets. Finally, we describe the concepts of expected return and risk and how to evaluate whether art is a good investment.

In this chapter we introduce probability on events. We follow an axiomatic approach that uses elementary set theory.

In Chapter 16, we defined an estimator of some unknown quantity, $\theta$, based on experimentally sampled data, $X$. This estimator, denoted by ${\widehat{\theta }}_{\text{ML}}(X)$, is called a maximum likelihood (ML) estimator, because it returns that value of $\theta$ that produces the highest likelihood of witnessing the particular sampled data.

Why does artistic activity tend to cluster in particular cities? This chapter seeks to explain how market size matters for cultural supply. We will explore the concentration of art and culture in urban centers by explaining the concept of economies of agglomeration. We delve deeper into the locational patterns in the visual arts, historical evidence of clustering of famous artists, and whether there is increasing concentration over time. Finally, we touch on economic impact studies of the arts.