introduction-probability.pdf

C compute the variance e f e f 2 of the random

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(c) Compute the variance E ( f - E f ) 2 of the random variable f ( k ) := k . Hint: Use a) and b). 9. Assume integrable, independent random variables f i : Ω R with E f 2 i < , mean E f i = m i and variance σ 2 i = E ( f i - m i ) 2 , for i = 1 , 2 , . . . , n. Compute the mean and the variance of (a) g = af 1 + b , where a, b R , (b) g = n i =1 f i .
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86 CHAPTER 4. EXERCISES 10. Let the random variables f and g be independent and Poisson dis- trubuted with parameters λ 1 and λ 2 , respectively. Show that f + g is Poisson distributed P f + g ( { k } ) = P ( f + g = k ) = k l =0 P ( f = l, g = k - l ) = . . . . Which parameter does this Poisson distribution have? 11. Assume that f ja g are independent random variables where E | f | < and E | g | < . Show that E | fg | < and E fg = E f E g. Hint: (a) Assume first f 0 and g 0 and show using the ”stair-case functions” to approximate f and g that E fg = E f E g. (b) Use (a) to show E | fg | < , and then Lebesgue’s Theorem for E fg = E f E g. in the general case. 12. Use H¨ older’s inequality to show Corollary 3.6.6. 13. Let f 1 , f 2 , . . . be non-negative random variables on (Ω , F , P ) . Show that E k =1 f k = k =1 E f k ( ≤ ∞ ) . 14. Use Minkowski’s inequality to show that for sequences of real numbers ( a n ) n =1 and ( b n ) n =1 and 1 p < it holds n =1 | a n + b n | p 1 p n =1 | a n | p 1 p + n =1 | b n | p 1 p . 15. Prove assertion (2) of Proposition 3.2.3. 16. Assume a sequence of i.i.d. random variables ( f k ) k =1 with E f 1 = m and variance E ( f 1 - m ) 2 = σ 2 . Use the Central Limit Theorem to show that P ω : f 1 + f 2 + · · · + f n - nm σ n x 1 2 π x -∞ e - u 2 2 du as n → ∞ .
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4.3. INTEGRATION 87 17. Let ([0 , 1] , B ([0 , 1]) , λ ) be a probability space. Define the functions f n : Ω R by f 2 n ( x ) := n 3 1I [0 , 1 / 2 n ] ( x ) and f 2 n - 1 ( x ) := n 3 1I [1 - 1 / 2 n, 1] ( x ) , where n = 1 , 2 , . . . . (a) Does there exist a random variable f : Ω R such that f n f almost surely? (b) Does there exist a random variable f : Ω R such that f n P f ? (c) Does there exist a random variable f : Ω R such that f n L P f ?
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Index λ -system, 31 lim inf n A n , 17 lim inf n a n , 17 lim sup n A n , 17 lim sup n a n , 17 π -system, 24 π -systems and uniqueness of mea- sures, 24 π - λ -Theorem, 31 σ -algebra, 10 σ -finite, 14 algebra, 10 axiom of choice, 32 Bayes’ formula, 19 binomial distribution, 25 Borel σ -algebra, 13 Borel σ -algebra on R n , 24 Carath´ eodory’s extension theorem, 21 central limit theorem, 75 change of variables, 58 Chebyshev’s inequality, 66 closed set, 12 conditional probability, 18 convergence almost surely, 71 convergence in L p , 71 convergence in distribution, 72 convergence in probability, 71 convexity, 66 counting measure, 15 Dirac measure, 15 distribution-function, 40 dominated convergence, 55 equivalence relation, 31 event, 10 existence of sets, which are not Borel, 32 expectation of a random variable, 47 expected value, 47 exponential distribution on R , 28 extended random variable, 60 Fubini’s Theorem, 61, 62 Gaussian distribution on R , 28 geometric distribution, 25 older’s inequality, 67 i.i.d. sequence, 74 independence of a family of random variables, 42 independence of a finite family of ran- dom variables, 42 independence of a sequence of events, 18
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