introduction-probability.pdf

10 show that in the definition of an algebra

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10. Show that in the definition of an algebra (Definition 1.1.1 one can replace (3’) A, B ∈ F implies that A B ∈ F by (3”) A, B ∈ F implies that A B ∈ F . 11. Prove that A \ B ∈ F if F is an algebra and A, B ∈ F . 12. Prove that i =1 A i ∈ F if F is a σ -algebra and A 1 , A 2 , ... ∈ F . 13. Give an example where the union of two σ -algebras is not a σ -algebra. 14. Let F be a σ -algebra and A ∈ F . Prove that G := { B A : B ∈ F} is a σ -algebra. 15. Prove that { B [ α, β ] : B ∈ B ( R ) } = σ { [ a, b ] : α a < b β } and that this σ -algebra is the smallest σ -algebra generated by the sub- sets A [ α, β ] which are open within [ α, β ]. The generated σ -algebra is denoted by B ([ α, β ]). 16. Show the equality σ ( G 2 ) = σ ( G 4 ) = σ ( G 0 ) in Proposition 1.1.8 holds. 17. Show that A ⊆ B implies that σ ( A ) σ ( B ). 18. Prove σ ( σ ( G )) = σ ( G ). 19. Let Ω = , A Ω, A = and F := 2 Ω . Define P ( B ) := 1 : B A = 0 : B A = . Is (Ω , F , P ) a probability space?
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4.1. PROBABILITY SPACES 79 20. Prove Proposition 1.2.6 (7). 21. Let (Ω , F , P ) be a probability space and A ∈ F such that P ( A ) > 0. Show that (Ω , F , μ ) is a probability space, where μ ( B ) := P ( B | A ) . 22. Let (Ω , F , P ) be a probability space. Show that if A, B ∈ F are inde- pendent this implies (a) A and B c are independent, (b) A c and B c are independent. 23. Let (Ω , F , P ) be the model of rolling two dice, i.e. Ω = { ( k, l ) : 1 k, l 6 } , F = 2 Ω , and P (( k, l )) = 1 36 for all ( k, l ) Ω . Assume A := { ( k, l ) : l = 1 , 2 or 5 } , B := { ( k, l ) : l = 4 , 5 or 6 } , C := { ( k, l ) : k + l = 9 } . Show that P ( A B C ) = P ( A ) P ( B ) P ( C ) . Are A, B, C independent ? 24. (a) Let Ω := { 1 , ..., 6 } , F := 2 Ω and P ( B ) := 1 6 card( B ). We define A := { 1 , 4 } and B := { 2 , 5 } . Are A and B independent? (b) Let Ω := { ( k, l ) : k, l = 1 , ..., 6 } , F := 2 Ω and P ( B ) := 1 36 card( B ). We define A := { ( k, l ) : k = 1 or k = 4 } and B := { ( k, l ) : l = 2 or l = 5 } . Are A and B independent? 25. In a certain community 60 % of the families own their own car, 30 % own their own home, and 20% own both (their own car and their own home). If a family is randomly chosen, what is the probability that this family owns a car or a house but not both? 26. Prove Bayes’ formula: Proposition 1.2.15. 27. Suppose we have 10 coins which are such that if the i th one is flipped then heads will appear with probability i 10 , i = 1 , 2 , . . . 10 . One chooses randomly one of the coins. What is the conditionally probability that one has chosen the fifth coin given it had shown heads after flipping? Hint: Use Bayes’ formula.
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80 CHAPTER 4. EXERCISES 28.* A class that is both, a π -system and a λ -system, is a σ -algebra. 29. Prove Property (3) in Lemma 1.4.4. 30.* Let (Ω , F , P ) be a probability space and assume A 1 , A 2 , ..., A n ∈ F are independent. Show that then A c 1 , A c 2 , ..., A c n are independent. 31. Let us play the following game: If we roll a die it shows each of the numbers { 1 , 2 , 3 , 4 , 5 , 6 } with probability 1 6 . Let k 1 , k 2 , ... ∈ { 1 , 2 , 3 , ... } be a given sequence. (a) First go: One has to roll the die k 1 times. If it did show all k 1 times 6 we won.
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