introduction-probability.pdf

2 assume that p 1 is a probability measure on f 1 and

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(2) Assume that P 1 is a probability measure on F 1 and define P 2 ( B 2 ) := P 1 ( { ω 1 Ω 1 : f ( ω 1 ) B 2 } ) . Then P 2 is a probability measure on F 2 . The proof is an exercise. Example 2.2.6 We want to simulate the flipping of an (unfair) coin by the random number generator: the random number generator of the computer gives us a number which has (a discrete) uniform distribution on [0 , 1]. So we take the probability space ([0 , 1] , B ([0 , 1]) , λ ) and define for p (0 , 1) the random variable f ( ω ) := 1I [0 ,p ) ( ω ) . Then it holds P 2 ( { 1 } ) := P 1 ( { ω Ω : f ( ω ) = 1 } ) = λ ([0 , p )) = p, P 2 ( { 0 } ) := P 1 ( { ω 1 Ω 1 : f ( ω 1 ) = 0 } ) = λ ([ p, 1]) = 1 - p. Assume the random number generator gives out the number x . If we would write a program such that ”output” = ”heads” in case x [0 , p ) and ”output” = ”tails” in case x [ p, 1], ”output” would simulate the flipping of an (unfair) coin, or in other words, ”output” has binomial distribution μ 1 ,p . Definition 2.2.7 [law of a random variable] Let (Ω , F , P ) be a prob- ability space and f : Ω R be a random variable. Then P f ( B ) := P ( { ω Ω : f ( ω ) B } ) is called the law or image measure of the random variable f .
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40 CHAPTER 2. RANDOM VARIABLES The law of a random variable is completely characterized by its distribution function which we introduce now. Definition 2.2.8 [distribution-function] Given a random variable f : Ω R on a probability space (Ω , F , P ), the function F f ( x ) := P ( { ω Ω : f ( ω ) x } ) is called distribution function of f . Proposition 2.2.9 [Properties of distribution-functions] The distribution-function F f : R [0 , 1] is a right-continuous non-decreasing function such that lim x →-∞ F ( x ) = 0 and lim x →∞ F ( x ) = 1 . Proof . (i) F is non-decreasing: given x 1 < x 2 one has that { ω Ω : f ( ω ) x 1 } ⊆ { ω Ω : f ( ω ) x 2 } and F ( x 1 ) = P ( { ω Ω : f ( ω ) x 1 } ) P ( { ω Ω : f ( ω ) x 2 } ) = F ( x 2 ) . (ii) F is right-continuous: let x R and x n x . Then F ( x ) = P ( { ω Ω : f ( ω ) x } ) = P n =1 { ω Ω : f ( ω ) x n } = lim n P ( { ω Ω : f ( ω ) x n } ) = lim n F ( x n ) . (iii) The properties lim x →-∞ F ( x ) = 0 and lim x →∞ F ( x ) = 1 are an exercise. Proposition 2.2.10 Assume that P 1 and P 2 are probability measures on B ( R ) and F 1 and F 2 are the corresponding distribution functions. Then the following assertions are equivalent: (1) P 1 = P 2 . (2) F 1 ( x ) = P 1 (( -∞ , x ]) = P 2 (( -∞ , x ]) = F 2 ( x ) for all x R .
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2.2. MEASURABLE MAPS 41 Proof . (1) (2) is of course trivial. We consider (2) (1): For sets of type A := ( a, b ] one can show that F 1 ( b ) - F 1 ( a ) = P 1 ( A ) = P 2 ( A ) = F 2 ( b ) - F 2 ( a ) . Now one can apply Proposition 1.2.23. Summary: Let (Ω , F ) be a measurable space and f : Ω R be a function. Then the following relations hold true: f - 1 ( A ) ∈ F for all A ∈ G where G is one of the systems given in Proposition 1.1.8 or any other system such that σ ( G ) = B ( R ) . Lemma 2 . 2 . 3 f is measurable: f - 1 ( A ) ∈ F for all A ∈ B ( R ) Proposition 2 . 2 . 2 f is a random variable i.e.
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