introduction-probability.pdf

There exist measurable step functions f n n 1 ie f n

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there exist measurable step functions ( f n ) n =1 i.e. f n = N n k =1 a n k 1I A n k with a n k R and A n k ∈ F such that f n ( ω ) f ( ω ) for all ω Ω as n → ∞ . Proposition 2 . 1 . 3 f - 1 (( a, b )) ∈ F for all - ∞ < a < b <
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42 CHAPTER 2. RANDOM VARIABLES 2.3 Independence Let us first start with the notion of a family of independent random variables. Definition 2.3.1 [independence of a family of random variables] Let (Ω , F , P ) be a probability space and f i : Ω R , i I , be random variables where I is a non-empty index-set. The family ( f i ) i I is called independent provided that for all distinct i 1 , ..., i n I , n = 1 , 2 , ... , and all B 1 , ..., B n ∈ B ( R ) one has that P ( f i 1 B 1 , ..., f i n B n ) = P ( f i 1 B 1 ) · · · P ( f i n B n ) . In case, we have a finite index set I , that means for example I = { 1 , ..., n } , then the definition above is equivalent to Definition 2.3.2 [independence of a finite family of random vari- ables] Let (Ω , F , P ) be a probability space and f i : Ω R , i = 1 , . . . , n , random variables. The random variables f 1 , . . . , f n are called independent provided that for all B 1 , ..., B n ∈ B ( R ) one has that P ( f 1 B 1 , ..., f n B n ) = P ( f 1 B 1 ) · · · P ( f n B n ) . The connection between the independence of random variables and of events is obvious: Proposition 2.3.3 Let , F , P ) be a probability space and f i : Ω R , i I , be random variables where I is a non-empty index-set. Then the following assertions are equivalent. (1) The family ( f i ) i I is independent. (2) For all families ( B i ) i I of Borel sets B i ∈ B ( R ) one has that the events ( { ω Ω : f i ( ω ) B i } ) i I are independent. Sometimes we need to group independent random variables. In this respect the following proposition turns out to be useful. For the following we say that g : R n R is Borel -measurable (or a Borel function) provided that g is ( B ( R n ) , B ( R ))-measurable. Proposition 2.3.4 [Grouping of independent random variables] Let f k : Ω R , k = 1 , 2 , 3 , ... be independent random variables. As- sume Borel functions g i : R n i R for i = 1 , 2 , ... and n i ∈ { 1 , 2 , ... } . Then the random variables g 1 ( f 1 ( ω ) , ..., f n 1 ( ω )) , g 2 ( f n 1 +1 ( ω ) , ..., f n 1 + n 2 ( ω )) , g 3 ( f n 1 + n 2 +1 ( ω ) , ..., f n 1 + n 2 + n 3 ( ω )) , ... are independent. The proof is an exercise.
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2.3. INDEPENDENCE 43 Proposition 2.3.5 [independence and product of laws] Assume that , F , P ) is a probability space and that f, g : Ω R are random variables with laws P f and P g and distribution-functions F f and F g , respectively. Then the following assertions are equivalent: (1) f and g are independent. (2) P (( f, g ) B ) = ( P f × P g )( B ) for all B ∈ B ( R 2 ) . (3) P ( f x, g y ) = F f ( x ) F g ( y ) for all x, y R . The proof is an exercise. Remark 2.3.6 Assume that there are Riemann-integrable functions p f , p g : R [0 , ) such that R p f ( x ) dx = R p g ( x ) dx = 1 , F f ( x ) = x -∞ p f ( y ) dy, and F g ( x ) = x -∞ p g ( y ) dy for all x R (one says that the distribution-functions F f and F g are abso- lutely continuous with densities p f and p g , respectively). Then the indepen- dence of f and g is also equivalent to the representation F ( f,g ) ( x, y ) = x -∞ y -∞ p f ( u ) p g ( v ) d ( u ) d ( v ) .
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