introduction-probability.pdf

Hence e f 2 k f 2 l e f 2 k e f 2 l c for k l

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Hence E f 2 k f 2 l = E f 2 k E f 2 l c for k = l . Consequently, E S 4 n nc + 3 n ( n - 1) c 3 cn 2 , and E n =1 S 4 n n 4 = n =1 E S 4 n n 4 n =1 3 c n 2 < . This implies that S 4 n n 4 a.s. 0 and therefore S n n a.s. 0. There are several strong laws of large numbers with other, in particular weaker, conditions. We close with a fundamental example concerning the convergence in distribution: the Central Limit Theorem (CLT). For this we need Definition 3.8.7 Let (Ω , F , P ) be a probability spaces. A sequence of I ndependent random variables f n : Ω R is called I dentically D istributed (i.i.d.) provided that the random variables f n have the same law, that means P ( f n λ ) = P ( f k λ ) for all n, k = 1 , 2 , ... and all λ R .
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3.8. MODES OF CONVERGENCE 75 Let (Ω , F , P ) be a probability space and ( f n ) n =1 be a sequence of i.i.d. ran- dom variables with E f 1 = 0 and E f 2 1 = σ 2 . By the law of large numbers we know f 1 + · · · + f n n P -→ 0 . Hence the law of the limit is the Dirac -measure δ 0 . Is there a right scaling factor c ( n ) such that f 1 + · · · + f n c ( n ) g, where g is a non-degenerate random variable in the sense that P g = δ 0 ? And in which sense does the convergence take place? The answer is the following Proposition 3.8.8 [Central Limit Theorem] Let ( f n ) n =1 be a sequence of i.i.d. random variables with E f 1 = 0 and E f 2 1 = σ 2 > 0 . Then P f 1 + · · · + f n σ n x 1 2 π x -∞ e - u 2 2 du for all x R as n → ∞ , that means that f 1 + · · · + f n σ n d g for any g with P ( g x ) = 1 2 π x -∞ e - u 2 2 du .
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76 CHAPTER 3. INTEGRATION
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Chapter 4 Exercises 4.1 Probability spaces 1. Prove that A ( B C ) = ( A B ) ( A C ). 2. Prove that ( i I A i ) c = i I A c i where A i Ω and I is an arbitrary index set. 3. Given a set Ω and two non-empty sets A, B Ω such that A B = . Give all elements of the smallest σ -algebra F on Ω which contains A and B . 4. Let x R . Is it true that { x } ∈ B ( R ), where { x } is the set, consisting of the element x only? 5. Assume that Q is the set of rational numbers. Is it true that Q ∈ B ( R )? 6. Given two dice with numbers { 1 , 2 , ..., 6 } . Assume that the probability that one die shows a certain number is 1 6 . What is the probability that the sum of the two dice is m ∈ { 1 , 2 , ..., 12 } ? 7. There are three students. Assuming that a year has 365 days, what is the probability that at least two of them have their birthday at the same day ? 8.* Definition: The system F ⊆ 2 Ω is a monotonic class , if (a) A 1 , A 2 , ... ∈ F , A 1 A 2 A 3 ⊆ · · · = n A n ∈ F , and (b) A 1 , A 2 , ... ∈ F , A 1 A 2 A 3 ⊇ · · · = n A n ∈ F . Show that if F ⊆ 2 Ω is an algebra and a monotonic class , then F is a σ -algebra. 9. Let E, F, G, be three events. Find expressions for the events that of E, F, G, 77
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78 CHAPTER 4. EXERCISES (a) only F occurs, (b) both E and F but not G occur, (c) at least one event occurs, (d) at least two events occur, (e) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur.
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