FinalPractice - 1 Dene symmetric dierence AB of two sets A...

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1. Define symmetric difference A Δ B of two sets A and B . Show that ( i =1 A i )Δ( i =1 B i ) ⊂ ∪ i =1 ( A i Δ B i ). Let (Ω , A , P ) be a probability space and E be a sub- σ -field of A . Con- sider F = { F A : P ( E Δ F ) = 0 for some E E } . Show that F is also a σ -field, containing E . Construct an example to argue that F may be strictly bigger than E . If X is an F -measurable random variable, show that there exists an E -measurable random variable Y such that X = Y a.s. [ P ]. (Hint: Start with indicator functions.) 2. Let μ be the Lebesgue measure. Show that μ ( A ) = 0 for any countable set A . If μ ( A ) = 0, is A necessarily countable? Prove or give a counterexample. If A contains an open interval, show that μ ( A ) > 0. If μ ( A ) = 0, show that A c is dense in R . 3. For any random variable X and any t, α > 0, show that P( X > t ) e - αt E( e αX ). Let X 1 , X 2 , . . . be an arbitrary sequence of random variables such that sup n 1 E( e αX n ) < for some α > 0. Show that, P( X n > C log n infinitely often) = 0 for some constant C > 0. You may use the fact that n =1 n - β < whenever β > 1.
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  • Spring '10
  • GHOSHAL
  • Probability, Probability theory, Xn, Lindeberg

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