This annex guides you through a sample of the Assessment Research Centre Online Testing System (ARCOTS) that you can access online using the code provided with this book. Detailed instructions are given here to take you through the assessment process. This will allow you to try out the use of online developmental assessment to plan teaching strategies for implementation in your classroom. We encourage you, when using the tests, to follow the CTT cycle discussed in Chapter 2, including planning, implementing and evaluating your teaching strategies.

ARCOTS overview

The ARCOTS tests were developed as part of the Assessment and Learning Partnerships Project (see the Introduction for more details). They are intended to be used by teachers as one of many sources of evidence that reflect student learning. They provide teachers with a means to track student progress for the purpose of informing and evaluating teaching. The test results are used to identify the point of learning readiness or zone of proximal development (ZPD) and they are reported as a level on the corresponding developmental progression. Three progressions were developed in the Assessment and Learning Partnerships Project for three learning domains: numeracy, reading comprehension and problem-solving; each domain has a corresponding test. The reports are available instantaneously, through ARCOTS, upon completion of an assessment. The progressions are discussed in Chapter 2.

Technical requirements

Table A2.1 lists the minimum technical requirements to successfully conduct ARCOTS tests.

ARCOTS can also be run on iPads using a Flash-compatible browser, such as Puffin, iSwifter or similar. If you are considering testing with iPads, we advise a trial run prior to commencing student testing to check compatibility and speed.

Note: please check whether you need an IT technician to whitelist (allow access to) the ARCOTS site in your school.

Test access

Samples of ARCOTS developmental assessments and the reporting system are provided on the website accompanying this book. The purpose of the sample tests is to give teachers an experience of using developmental progressions to inform teaching and implement some of the processes described in this book.

Before the testing and reporting systems can be accessed you will need to register with the code and the link provided on the companion website. To register you will need to have a valid email address.

Introduction

This chapter will provide you with the fundamental skills for working with children, young people and their families to promote mental health. Adolescent mental health and wellbeing are covered in more detail elsewhere in this text, but it is important to acknowledge that good mental health is an important goal from birth (and even before birth), throughout childhood and into adolescence. Some adolescent mental health disorders are adolescent limited – that is, they begin and end in the period of adolescence – but mostly mental health and mental health disorders are experienced along a continuum. Indeed, as you will learn from this chapter, temporal assessment of children's behaviours is paramount in any assessment of mental health. As children mature, their behaviours change accordingly. Behaviours that can be expected among young children should disappear as they mature, such as the inability to regulate emotions, or to experience empathy for others. It is therefore imperative that mental health and behavioural assessments are not completed in one encounter. Paediatric mental health clinicians view the child or young person as a representative of a family system. This means that the family is necessarily incorporated into every facet of care, including assessment, planning, treatment and evaluation of care. Children and young people, their parents and guardians, families, school and broader communities are all considered in the planning for optimal mental healthcare service delivery.

Children and young people are actually at high risk of mental health disorders, and in Australia 8.3 per cent of children and young people live with mild, moderate or severe mental disorders (Lawrence et al., 2016). Since the first Australian survey of children's mental health was reported in 2000, the prevalence of children's mental health disorders has been stable. The National Youth Mental Health Initiative launched by the Australian Government in 2005 (Headspace) has focused on increasing prevention and treatment services for children and young people. It seems that for children identified as having mental health disorders, the uptake of services is high. But there continues to be a need for improving the prevention and early intervention strategies and services, especially for those children and young people already experiencing mental health problems.

Why do so many students find probability difficult? Even the most mathematically competent often find probability a subject that is difficult to use and understand. The difficulty stems from the fact that most problems in probability, even ones that are easy to understand, cannot be solved by using cookbook recipes as is sometimes the case in other areas of mathematics. Instead, each new problem often requires imagination and creative thinking. That is why probability is difficult, but also why it is fun and engaging. Probability is a fascinating subject and I hope, in this book, to share my enthusiasm for the subject.

Probability is best taught to beginning students by using motivating examples and problems, and a solution approach that gives the students confidence to solve problems on their own. The examples and problems should be relevant, clear, and instructive. This book is not written in a theorem–proof style, but proofs flow with the subsequent text and no mathematics is introduced without specific examples and applications to motivate the theory. It distinguishes itself from other introductory probability texts by its emphasis on why probability is so relevant and how to apply it. Every attempt has been made to create a student-friendly book and to help students understand what they are learning, not just to learn it.

This textbook is designed for a first course in probability at an undergraduate level or first-year graduate level. It covers all of the standard material for such courses, but it also contains many topics that are usually not found in introductory probability books – such as stochastic simulation. The emphasis throughout the book is on probability, but attention is also given to statistics. In particular, Bayesian inference is discussed at length and illustrated with several illuminating examples. The book can be used in a variety of disciplines, ranging from applied mathematics and statistics to computer science, operations research, and engineering, and is suitable not only for introductory courses, but also for self-study. The prerequisite knowledge is a basic course in calculus.

Good problems are an essential part of each textbook. The field of probability is well known for being a subject that can best be acquired by the process of learning-by-doing. Much care has been taken to present problems that will enhance the student's understanding of probability.

Many probability problems require counting techniques. In particular, these techniques are extremely useful for computing probabilities in a chance experiment in which all possible outcomes are equally likely. In such experiments, one needs effective methods to count the number of outcomes in any specific event. In counting problems, it is important to know whether the order in which the elements are counted is relevant or not.

In the discussion below, we use the fundamental principle of counting: if there are a ways to do one activity and b ways to do another activity, then there are a × b ways of doing both. As an example, suppose that you go to a restaurant to get some breakfast. The menu says pancakes, waffles, or fried eggs, while for a drink you can choose between juice, coffee, tea, and hot chocolate. Then the total number of different choices of food and drink is 3 × 4 = 12. As another example, how many different license plates are possible when the license plate displays a nonzero digit, followed by three letters, followed by three digits? The answer is 9×26×26×26×10×10× 10 = 158,184,000 license plates.

Permutations

How many different ways can you arrange a number of different objects such as letters or numbers? For example, what is the number of different ways that the three letters A, B, and C can be arranged? By writing out all the possibilities ABC, ACB, BAC, BCA, CAB, and CBA, you can see that the total number is 6. This brute-force method of writing down all the possibilities and counting them is naturally not practical when the number of possibilities gets large, for example the number of different ways to arrange the 26 letters of the alphabet. You can also determine that the three letters A, B, and C can be written down in six different ways by reasoning as follows. For the first position there are three available letters to choose from, for the second position there are two letters over to choose from, and only one letter for the third position. Therefore the total number of possibilities is 3 × 2 × 1 = 6. The general rule should now be evident. Suppose that you have n distinguishable objects.

Introduction

This book is about a country's financial and monetary regime and the interaction between the regime and the economy. The financial and monetary regime is an important element of the economic and political environment in which we live and work, and some basic knowledge of it is necessary if one wants to consider oneself educated. In fact, anyone not familiar with the basic elements of the financial and monetary regime and its relationship to economic activity should consider him- /herself less than well informed, both as an individual and as a member of society. Lacking knowledge about its basic elements is not only dangerous to your economic health but dangerous to your ability to participate in the political process.

Knowledge of the financial and monetary regime will not guarantee economic success, but it will help you avoid mistakes that will surely limit your lifetime wealth.On a broader level, lacking knowledge about the basic elements of the financial and monetary regime renders you a low-information voter, or “useful idiot”, easily manipulated by politicians on either side of the aisle. The term gained new life in late 2014 when it became widely known that one of the major consultants to the 2010 Affordable Care Act claimed, to a group of economists at a conference, that the public's lack of economic understanding and their basic “stupidity” about economics played an important role in enacting a major overhaul of and expanded role of government in the U.S. health system (Bierman, 2014).

This was a dark day for the role of an economist in public policy, but it offers an important lesson. Irrespective of one's view of the Act, the mindset that subterfuge is acceptable for major expansions in government should give everyone pause about government activism. Government, whether to the right or left, does not always have the best interest of the individual in mind and often relies on uninformed voters to pass complex legislation and pursue policies that may not be in the best interests of the country. At a minimum, understanding the basic elements of the financial and monetary regime will help you manage your wealth and render you a more informed observer of important public policy debates that greatly influence your life and reduce your reliance on “talking heads”, who dominate the news media and who, unfortunately, haven't a clue about most of the contents of this book.

In Chapter 2, conditional probabilities were introduced by conditioning upon the occurrence of an event B of nonzero probability. In applications, this event B is often of the form Y = b for a discrete random variable Y. However, when the random variable Y is continuous, the condition Y = b has probability zero for any number b. In this chapter we will develop techniques for handling a condition provided by the observed value of a continuous random variable. You will see that the conditional probability density function of X given Y = b for continuous random variables is analogous to the conditional probability mass function of X given Y = b for discrete random variables. The conditional distribution of X given Y = b enables us to define the natural concept of conditional expectation of X given Y = b. This concept allows for an intuitive understanding and is of utmost importance. In statistical applications, it is often more convenient to work with conditional expectations instead of the correlation coefficient when measuring the strength of the relationship between two dependent random variables. In applied probability problems, the computation of the expected value of a random variable X is often greatly simplified by conditioning on an appropriately chosen random variable Y. Learning the value of Y provides additional information about the random variable X and for that reason the computation of the conditional expectation of X given Y = b is often simple. Much attention will be paid to the law of conditional probability and the law of conditional expectation. These laws are extremely useful when solving applied probability problems. Among other things, they will be used in the solving of stochastic optimization problems. In the final section we explain Bayesian inference for continuous models and give several statistical applications.

Conditional Distributions

Suppose that the random variables X and Y are defined on the same sample space _ with probability measure P. A basic question for dependent random variables X and Y is: if the observed value of Y is y, what distribution now describes the distribution of X? We first answer this question for the discrete case.

In many practical applications of probability, physical situations are better described by random variables that can take on a continuum of possible values rather than a discrete number of values. Examples are the decay time of a radioactive particle, the time until the occurrence of the next earthquake in a certain region, the lifetime of a battery, the annual rainfall in London, electricity consumption in kilowatt hours, and so on. These examples make clear what the fundamental difference is between discrete random variables taking on a discrete number of values and continuous random variables taking on a continuum of values. Whereas a discrete random variable associates positive probabilities with its individual values, any individual value has probability zero for a continuous random variable. It is only meaningful to speak of the probability of a continuous random variable taking on a value in some interval. Taking the lifetime of a battery as an example, it will be intuitively clear that the probability of this lifetime taking on a specific value becomes zero when a finer and finer unit of time is used. If you can measure the heights of people with infinite precision, the height of a randomly chosen person is a continuous random variable. In reality, heights cannot be measured with infinite precision, but the mathematical analysis of the distribution of people's heights is greatly simplified when using a mathematical model in which the height of a randomly chosen person is modeled as a continuous random variable. Integral calculus is required to formulate the continuous analogue of a probability mass function of a discrete random variable.

The first purpose of this chapter is to familiarize you with the concept of the probability density of a continuous random variable. This is always a difficult concept for the beginning student. However, integral calculus enables us to give an enlightening interpretation of probability density. The second purpose of this chapter is to introduce you to important probability densities such as the uniform, exponential, gamma, Weibull, beta, normal, and lognormal densities among others. In particular, the exponential and normal distributions are treated in depth. Many practical phenomena can be modeled by these distributions, which are of fundamental importance. Many examples are given to illustrate this.

Introduction

Government regulation and supervision of the financial system constitute one of the three components of the nation's financial and monetary regime. Chapter 8 discussed how government regulation and supervision evolved over time as the nation's monetary standards shifted from commodity to representative commodity and, finally, to fiat-based standards. That discussion was designed to explain how government emerged as a major force in the nation's financial and monetary regime. The discussion in Chapter 8 was necessarily broad. This chapter is more focused, in two ways: First, the rationale for government regulation and supervision is expanded to incorporate different perspective of market failure; and, second, the important and specific types of government regulation and supervision are outlined.

Asymmetric Information, Adverse Selection and Lemons

The rationale for government regulation and supervision can now be extended to incorporate two other types of market failure that interfere with the ability of the nation's financial and monetary regime to operate smoothly. The two new perspectives of market failure are referred to as the asymmetric information and adverse selection problems. These are general problems in any market, and can be introduced by considering a market for a commodity such as a used car or home. George Akerlof (1970) is credited with bringing attention to how asymmetric information and adverse selection render the used car market less inefficient.His analysis is now referred to the “lemon problem” in the used car market. The principles, however, are general.

Asymmetric information refers to the fact the buyer and the seller of the used car do not have the same information set. The seller generally has more information about the car than the buyer. The seller knows the car engine burns too much oil or has a weak transmission even though the car drives well for short distances. The buyer does not have this knowledge, and, while the buyer can pay to have a mechanic examine the car, this is expensive and time-consuming and may not provide reliable information if the defect is subtle. That is, the seller knows the car is a lemon but the potential buyer does not. As a result, the potential buyer assumes the used car is of average quality irrespective of how well it is represented and, hence, will offer a price based on that assumption alone.