introduction-probability.pdf

Step 4 is it possible to describe the distribution of

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Step 4: Is it possible to describe the distribution of the values f may take? Or before, what do we mean by a distribution? Some basic distributions are discussed in Section 1.3. Step 5: What is a good method to estimate E f ? We can take a sequence of independent (take this intuitive for the moment) random variables f 1 , f 2 , ... , having the same distribution as f , and expect that 1 n n i =1 f i ( ω ) and E f are close to each other. This lieds us to the Strong Law of Large Numbers discussed in Section 3.8. Notation. Given a set Ω and subsets A, B Ω, then the following notation is used: intersection: A B = { ω Ω : ω A and ω B } union: A B = { ω Ω : ω A or (or both) ω B } set-theoretical minus: A \ B = { ω Ω : ω A and ω B } complement: A c = { ω Ω : ω A } empty set: = set, without any element real numbers: R natural numbers: N = { 1 , 2 , 3 , ... } rational numbers: Q indicator-function: 1I A ( x ) = 1 if x A 0 if x A Given real numbers α, β , we use α β := min { α, β } .
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8 CONTENTS
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Chapter 1 Probability spaces In this chapter we introduce the probability space, the fundamental notion of probability theory. A probability space (Ω , F , P ) consists of three compo- nents. (1) The elementary events or states ω which are collected in a non-empty set Ω. Example 1.0.1 ( a ) If we roll a die, then all possible outcomes are the numbers between 1 and 6. That means Ω = { 1 , 2 , 3 , 4 , 5 , 6 } . ( b ) If we flip a coin, then we have either ”heads” or ”tails” on top, that means Ω = { H, T } . If we have two coins, then Ω = { ( H, H ) , ( H, T ) , ( T, H ) , ( T, T ) } is the set of all possible outcomes. ( c ) For the lifetime of a bulb in hours we can choose Ω = [0 , ) . (2) A σ -algebra F , which is the system of observable subsets or events A Ω. The interpretation is that one can usually not decide whether a system is in the particular state ω Ω, but one can decide whether ω A or ω A . (3) A measure P , which gives a probability to all A ∈ F . This probability is a number P ( A ) [0 , 1] that describes how likely it is that the event A occurs. For the formal mathematical approach we proceed in two steps: in the first step we define the σ -algebras F , here we do not need any measure. In the second step we introduce the measures. 9
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10 CHAPTER 1. PROBABILITY SPACES 1.1 Definition of σ -algebras The σ -algebra is a basic tool in probability theory. It is the set the proba- bility measures are defined on. Without this notion it would be impossible to consider the fundamental Lebesgue measure on the interval [0 , 1] or to consider Gaussian measures, without which many parts of mathematics can not live. Definition 1.1.1 [ σ -algebra, algebra, measurable space] Let Ω be a non-empty set. A system F of subsets A Ω is called σ - algebra on Ω if (1) , Ω ∈ F , (2) A ∈ F implies that A c := Ω \ A ∈ F , (3) A 1 , A 2 , ... ∈ F implies that i =1 A i ∈ F .
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