introduction-probability.pdf

B second go we roll k 2 times again we win if it

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(b) Second go: We roll k 2 times. Again, we win if it would all k 2 times show 6. And so on... Show that (a) The probability of winning infinitely many often is 1 if and only if n =1 1 6 k n = , (b) The probability of loosing infinitely many often is always 1. Hint: Use the lemma of Borel-Cantelli. 32. Let n 1, k ∈ { 0 , 1 , ..., n } and n k := n ! k !( n - k )! where 0! := 1 and k ! := 1 · 2 · · · k , for k 1. Prove that one has ( n k ) possibilities to choose k elements out of n ele- ments. 33. Binomial distributon: Assume 0 < p < 1, Ω := { 0 , 1 , 2 , ..., n } , n 1, F := 2 Ω and μ n,p ( B ) := k B n k p n - k (1 - p ) k . (a) Is (Ω , F , μ n,p ) a probability space? (b) Compute max k =0 ,...,n μ n,p ( { k } ) for p = 1 2 . 34. Geometric distribution: Let 0 < p < 1, Ω := { 0 , 1 , 2 , ... } , F := 2 Ω and μ p ( B ) := k B p (1 - p ) k .
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4.2. RANDOM VARIABLES 81 (a) Is (Ω , F , μ p ) a probability space? (b) Compute μ p ( { 0 , 2 , 4 , 6 , ... } ). 35. Poisson distribution: Let λ > 0, Ω := { 0 , 1 , 2 , ... } , F := 2 Ω and π λ ( B ) := k B e - λ λ k k ! . (a) Is (Ω , F , π λ ) a probability space? (b) Compute k Ω λ ( { k } ). 4.2 Random variables 1. Let A 1 , A 2 , ... Ω. Show that (a) lim inf n →∞ 1I A n ( ω ) = 1I lim inf n →∞ A n ( ω ), (b) lim sup n →∞ 1I A n ( ω ) = 1I lim sup n →∞ A n ( ω ), for all ω Ω. 2. Let (Ω , F , P ) be a probability space and A Ω a set. Show that A ∈ F if and only if 1I A : Ω R is a random variable. 3. Show the assertions (1), (3) and (4) of Proposition 2.1.5. 4. Let (Ω , F , P ) be a probability space and f : Ω R . (a) Show that, if F = {∅ , Ω } , then f is measurable if and only if f is constant. (b) Show that, if P ( A ) is 0 or 1 for every A ∈ F and f is measurable, then P ( { ω : f ( ω ) = c } ) = 1 for a constant c. 5. Let (Ω , F ) be a measurable space and f n , n = 1 , 2 , . . . a sequence of random variables. Show that lim inf n →∞ f n and lim sup n →∞ f n are random variables. 6. Complete the proof of Proposition 2.2.9 by showing that for a distribu- tion function F g ( x ) = P ( { ω : g x } ) of a random variable g it holds lim x →-∞ F g ( x ) = 0 and lim x →∞ F g ( x ) = 1 .
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82 CHAPTER 4. EXERCISES 7. Let f : Ω R be a map. Show that for A 1 , A 2 , · · · ⊆ R it holds f - 1 i =1 A i = i =1 f - 1 ( A i ) . 8. Let (Ω 1 , F 1 ), (Ω 2 , F 2 ), (Ω 3 , F 3 ) be measurable spaces. Assume that f : Ω 1 Ω 2 is ( F 1 , F 2 )-measurable and that g : Ω 2 Ω 3 is ( F 2 , F 3 )- measurable. Show that then g f : Ω 1 Ω 3 defined by ( g f )( ω 1 ) := g ( f ( ω 1 )) is ( F 1 , F 3 )-measurable. 9. Prove Proposition 2.1.4. 10. Let (Ω , F , P ) be a probability space, ( M, Σ) a measurable space and assume that f : Ω M is ( F , Σ)-measurable. Show that μ with μ ( B ) := P ( { ω : f ( ω ) B } ) for B Σ is a probability measure on Σ. 11. Assume the probability space (Ω , F , P ) and let f, g : Ω R be mea- surable step-functions. Show that f and g are independent if and only if for all x, y R it holds P ( { f = x, g = y } ) = P ( { f = x } ) P ( { g = y } ) 12. Assume the product space ([0 , 1] × [0 , 1] , B ([0 , 1]) ⊗B ([0 , 1]) , λ × λ ) and the random variables f ( x, y ) := 1I [0 ,p ) ( x ) and g ( x, y ) := 1I [0 ,p ) ( y ), where 0 < p < 1. Show that (a) f and g are independent, (b) the law of f + g , P f + g ( { k } ) , for k = 0 , 1 , 2 is the binomial distri- bution μ 2 ,p .
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