midterm-sol

midterm-sol - Mathematics 136, Midterm 2008 Write your name...

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Mathematics 136, Midterm 2008 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use three pages (2 sides each) of notes. You have 90 minutes to solve all three questions, each worth points as marked (maximum of 50). Complete reasoning is required for full credit. You may cite lecture notes and homework sets, as needed, stating precisely the result you use, why and how it applies. 1. (4x4) Provide the complete, accurate and rigorous deFnition of four of the following concepts (it is not enough to merely cite their location in text). a) A measurable space (Ω , F ), the structure of F , and a probability measure P on it. ANS: F is a σ -Feld of subsets of Ω and detail DeFnitions 1.1.1 and 1.1.2. b) The law of a random variable X , its distribution function and the convergence in law of random variables X n to a random variable X . ANS: Detail DeFnitions 1.4.1, 1.4.5 and 1.4.10. c) The conditional expectation of X L 2 , F , P ) given a random variable Y on (Ω , F ). ANS: Detail DeFnition 2.1.3. d) A stochastic process { X t , t IR } , its Fnite dimensional distributions and the conditions which make them consistent . ANS: Detail DeFnitions 3.1.1, 3.1.5 and 3.1.13. e) A Gaussian stochastic process { X t } and its auto-covariance and mean functions ρ ( t, s ) and μ ( t ). ANS: Detail DeFnition 3.2.17 and ideally also DeFnition 3.2.8 for a Gaussian stochastic process X t . Then, μ ( t ) = E X t and ρ ( t, s ) = E [( X t - μ ( t )( X s - μ ( s ))]. 2. (2+3+2+3+3+3) a) Suppose N is a random variable that is uniformly distributed in the Fnite set { 1 , 2 , . . . , m } for some non-random Fnite integer m . ±ixing 0 < p < 1 let { ξ k } be independent and identically distributed random variables on the same probability space (Ω , F , P ), that are independent of N and such that p = P ( ξ k = 1) = 1 - P ( ξ k = - 1). Explain why for any non-random constant r , Y ( ω ) = N ( ω ) X k =1 (1 + k ( ω )) , can be written as Y = m k =1 (1 + k ) I { N k } and why it is in L 1 , F , P ). ANS: Applying the identity g ( n, a 1 , . . . , a m ) = n k =1 a k = m k =1 a k 1 n k which is valid for any a k IR 1
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and positive integers n m we get the alternative formula for Y . Next Y = g ( N, 1 + 1 , . . . , 1 +
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This note was uploaded on 11/08/2009 for the course STAT 219 taught by Professor -2 during the Fall '08 term at Stanford.

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midterm-sol - Mathematics 136, Midterm 2008 Write your name...

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