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Unformatted text preview: 1. Define symmetric difference A B of two sets A and B . Show that ( i =1 A i )( i =1 B i ) i =1 ( A i B i ). Let ( , A ,P ) be a probability space and E be a sub field of A . Con sider F = { F A : P ( E F ) = 0 for some E E } . Show that F is also a field, containing E . Construct an example to argue that F may be strictly bigger than E . If X is an Fmeasurable random variable, show that there exists an Emeasurable random variable Y such that X = Y a.s. [ P ]. (Hint: Start with indicator functions.) 2. Let be the Lebesgue measure. Show that ( A ) = 0 for any countable set A . If ( A ) = 0, is A necessarily countable? Prove or give a counterexample. If A contains an open interval, show that ( A ) > 0. If ( A ) = 0, show that A c is dense in R . 3. For any random variable X and any t, > 0, show that P( X > t ) e t E( e X ). Let X 1 ,X 2 ,... be an arbitrary sequence of random variables such that sup n 1 E( e X n ) < for some > 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 < whenever > 1....
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This note was uploaded on 12/11/2010 for the course STAT 779 taught by Professor Ghoshal during the Spring '10 term at N.C. State.
 Spring '10
 GHOSHAL
 Probability

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