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# lectures - MATH 235A Probability Theory Lecture Notes Fall...

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MATH 235A – Probability Theory Lecture Notes, Fall 2009 Part I: Foundations Dan Romik Department of Mathematics, UC Davis Draft version of 10/14/2009 Lecture 1: Introduction 1.1 What is probability theory? In this course we’ll learn about probability theory. But what exactly is probability theory? Like some other mathematical fields (but unlike some others), it has a dual role: It is a rigorous mathematical theory – with definitions, lemmas, theorems, proofs etc. It is a mathematical model that purports to explain or model real-life phenomena. We will concentrate on the rigorous mathematical aspects, but we will try not to forget the connections to the intuitive notion of real-life probability. These connections will enhance our intuition, and they make probability an extremely useful tool in all the sciences. And they make the study of probability much more fun , too! A note of caution is in order, though: Mathematical models are only as good as the assumptions they are based on. So probability can be used , and it can be (and quite frequently is) ab used... 1

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1.2 The algebra of events A central notion in probability is that of the algebra of events (we’ll clarify later what the word “algebra” means in this context). We begin with an informal discussion. We imagine that probability is a function, denoted P , that takes as its argument an “event” (i.e., occurrence of something in a real-life situation involving uncertainty) and returns a number in [0 , 1] representing how likely this event is to occur. For example, if a fair coin is tossed 10 times and we denote the results of the tosses by X 1 , X 2 , . . . , X 10 (where each of X i is 0 or 1, signifying “tails” or “heads”), then we can write statements like P ( X i = 0) = 1 / 2 , (1 i 10) , P 10 X i =1 X i = 4 ! = ( 10 4 ) 2 10 . Note that if A and B represent events (meaning, for the purposes of the present informal discussion, objects that have a well-defined probability), then we expect that the phrases A did not occur”, “ A and B both occurred” and “at least one of A and B occurred” also represent events. We can denote these new events by ¬ A , A B , and A B , respectively. Thus, the set of events is not just a set – it is a set with some extra structure, namely the ability to perform negation , conjunction and disjunction operations on its elements. Such a set is called an algebra in some contexts. But what if the coin is tossed an infinite number of times? In other words, we now imagine an infinite sequence X 1 , X 2 , X 3 , . . . of (independent) coin toss results. We want to be able to ask questions such as P (infinitely many of the X i ’s are 0) = ? P lim n →∞ 1 n n X k =0 X k = 1 2 ! = ? P X k =0 2 X k - 1 k converges ! = ? Do such questions make sense? (And if they do, can you guess what the answers are?) Maybe it is not enough to have an informal discussion to answer this...
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lectures - MATH 235A Probability Theory Lecture Notes Fall...

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