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According to Leo Breiman (1968), probability theory has a right and a left hand. The right hand refers to rigorous mathematics, and the left hand refers to ‘pro- bilistic thinking’. The combination of these two aspects makes probability theory one of the most exciting ?elds in mathematics. One can study probability as a purely mathematical enterprise, but even when you do that, all the concepts that arisedo haveameaningontheintuitivelevel.Forinstance,wehaveto de?newhat we mean exactly by independent events as a mathematical concept, but clearly, we all know that when we ?ip a coin twice, the event that the ?rst gives heads is independent of the event that the second gives tails. Why have I written this book? I have been teaching probability for more than ?fteen years now, and decided to do something with this experience. There are already many introductory texts about probability, and there had better be a good reason to write a new one. I will try to explain my reasons now.



Chapter 1. Experiments

We start our investigations with a number of elementary examples which involve probability. These examples lead to a definition of an experiment, which consists of a space of possible outcomes, together with an assignment of probabilities to each of these outcomes. We define and study basic concepts associated with experiments, including combinatorics, independence, conditional probabilities and a first law of large numbers.

Chapter 2. Random Variables and Random Vectors

It often happens that we do not really care about the outcome of an experiment itself, but rather we are interested in some consequence of this outcome. For instance, a gambler is not primarily interested in the question whether or not heads comes up, but instead in the financial consequences of such an outcome. Hence the gambler is interested in a function of the outcome, rather than in the outcome itself. Such a function is called a random variable and in this chapter we define and study such random variables.

Chapter 3. Random Walk

In this chapter, we concentrate on an experiment that is strongly related to coin flips, the random walk. We shall prove a number of results that are very surprising and counterintuitive. It is quite nice that such results can be proved at this point already. The proofs are all based on (ingenious) counting methods.

Chapter 4. Limit Theorems

We have already encountered a number of limit theorems. For instance, the law of large numbers Theorem 1.6.1 and the arc-sine law Theorem 3.2.3 were statements about the limiting behaviour of a growing number of random variables. In this chapter, we state new laws of large numbers, and a primitive version of the famous central limit theorem.

Chapter I. Intermezzo

So far we have been dealing with countable sample spaces. The reason for this, as mentioned before, is that we can develop many interesting probabilistic notions without needing too many technical details. However, there are a number of intuitive probabilistic ideas which can not be studied using only countable sample spaces. Perhaps the most obvious kind of things which can not be captured in the countable setting are those involving an infinitely fine operation, such as choosing a random point from a line segment. In this intermezzo we see why this is so, and observe some problems that arise from this fact. After the intermezzo, we continue with the study of continuous probability theory, which does capture such infinitely fine operations.

Chapter 5. Continuous Random Variables and Vectors

We have seen in the Intermezzo that there is a need to generalise the notions of an experiment and a random variable. In this chapter we suggest a set up which allows us to do this. As in the first two chapters, we first define experiments, and after that random variables. The theory in this chapter is built up similarly as the theory in the first two chapters.

Chapter 6. Infinitely Many Repetitions

In the Intermezzo, we found that an infinitely fine operation like choosing a point on a line segment can not be captured with countable sample spaces, and hence we extended our notion of a sample space as to be able to deal with that kind of operations.

Chapter 7. The Poisson Process

In this chapter we discuss a probabilistic model which can be used to describe the occurrences of unpredictable events, which do exhibit a certain amount of statistical regularity. Examples to keep in mind are the moments at which telephone calls are received in a call centre, the moments at which customers enter a particular shop, or the moments at which California is hit by an earthquake. We refer to an occurrence of such an unpredictable event simply as an occurrence.

Chapter 8. Limit Theorems

In this chapter, we will be concerned with some more general limit theorems. In particular, we shall generalise the central limit Theorem 4.2.1. The method of proof will also lead to a new formulation of the law of large numbers. The methods behind these results are not so easy. They rely on concepts from complex analysis. To make sure that that you know what we are talking about, in Section 8.2 there will be a short introduction to complex analysis which contains all the background necessary for the development in this chapter. In this chapter, random variables can be discrete or continuous. We start by formalising a mode of convergence that we have, in fact, already seen.

Chapter 9. Extending the Probabilities

In this chapter we discuss how to extend the collection of events in rather general situations. This small chapter is the bridge between probability without measure theory and probability with measure theory.


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