introduction-probability.pdf

Now we get that 0 n i 1 α i 1i a i n i 1 j i i α i

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Now we get that 0 = n i =1 α i 1I A i = n i =1 j I i α i 1I C j = N j =1 i : j I i α i 1I C j = N j =1 γ j 1I C j so that γ j = 0 if C j = . From this we get that n i =1 α i P ( A i ) = n i =1 j I i α i P ( C j ) = N j =1 i : j I i α i P ( C j ) = N j =1 γ j P ( C j ) = 0 . Proposition 3.1.3 Let , F , P ) be a probability space and f, g : Ω R be measurable step-functions. Given α, β R one has that E ( αf + βg ) = α E f + β E g. Proof . The proof follows immediately from Lemma 3.1.2 and the definition of the expected value of a step-function since, for f = n i =1 α i 1I A i and g = m j =1 β j 1I B j , one has that αf + βg = α n i =1 α i 1I A i + β m j =1 β j 1I B j and E ( αf + βg ) = α n i =1 α i P ( A i ) + β m j =1 β j P ( B j ) = α E f + β E g.
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3.1. DEFINITION OF THE EXPECTED VALUE 47 Definition 3.1.4 [step two, f is non-negative] Given a probability space (Ω , F , P ) and a random variable f : Ω R with f ( ω ) 0 for all ω Ω. Then E f = Ω fd P = Ω f ( ω ) d P ( ω ) := sup { E g : 0 g ( ω ) f ( ω ) , g is a measurable step-function } . Note that in this definition the case E f = is allowed. In the last step we will define the expectation for a general random variable. To this end we decompose a random variable f : Ω R into its positive and negative part f ( ω ) = f + ( ω ) - f - ( ω ) with f + ( ω ) := max { f ( ω ) , 0 } ≥ 0 and f - ( ω ) := max {- f ( ω ) , 0 } ≥ 0 . Definition 3.1.5 [step three, f is general] Let (Ω , F , P ) be a proba- bility space and f : Ω R be a random variable. (1) If E f + < or E f - < , then we say that the expected value of f exists and set E f := E f + - E f - [ -∞ , ] . (2) The random variable f is called integrable provided that E f + < and E f - < . (3) If the expected value of f exists and A ∈ F , then A fd P = A f ( ω ) d P ( ω ) := Ω f ( ω )1I A ( ω ) d P ( ω ) . The expression E f is called expectation or expected value of the random variable f . Remark 3.1.6 The fundamental Lebesgue -integral on the real line can be introduced by the means, we have to our disposal so far, as follows: Assume a function f : R R which is ( B ( R ) , B ( R ))-measurable. Let f n : ( n - 1 , n ] R be the restriction of f which is a random variable with respect to B (( n - 1 , n ]) = B ( n - 1 ,n ] . In Section 1.3.4 we have introduced the Lebesgue measure λ = λ n on ( n - 1 , n ]. Assume that f n is integrable on ( n - 1 , n ] for all n = 1 , 2 , ... and that n = -∞ ( n - 1 ,n ] | f ( x ) | ( x ) < ,
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48 CHAPTER 3. INTEGRATION then f : R R is called integrable with respect to the Lebesgue -measure on the real line and the Lebesgue -integral is defined by R f ( x ) ( x ) := n = -∞ ( n - 1 ,n ] f ( x ) ( x ) . Now we go the opposite way: Given a Borel set I and a map f : I R which is ( B I , B ( R )) measurable, we can extend f to a ( B ( R ) , B ( R ))-measurable function f by f ( x ) := f ( x ) if x I and f ( x ) := 0 if x I . If f is Lebesgue -integrable, then we define I f ( x ) ( x ) := R f ( x ) ( x ) . Example 3.1.7 A basic example for our integration is as follows: Let Ω = { ω 1 , ω 2 , ... } , F := 2 Ω , and P ( { ω n } ) = q n [0 , 1] with n =1 q n = 1. Given f : Ω R we get that f is integrable if and only if n =1 | f ( ω n ) | q n < , and the expected value exists if either { n : f ( ω n ) 0 } f ( ω n ) q n < or { n : f ( ω n ) 0 } ( - f ( ω n )) q n < .
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