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Unformatted text preview: Mathematics 136, Midterm 2007 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 definition of four of the following concepts. a) A random variable X defined on ( , F , P ) and its law P X . ANS: Detail Definitions 1.2.1 and 1.4.1. b) The space L 2 ( , F , P ) and the convergence in 2mean of random variables X n to a random variable X . ANS: Detail the introduction to Section 1.3.2 and Definition 1.3.17. c) The conditional expectation of X L 1 ( , F , P ) given a field G F . ANS: Detail Definition 2.1.5. d) The characteristic function of a random vector X with values in IR n , and a Gaussian random vector Y with values in IR n . ANS: Detail Definitions 3.2.1 and 3.2.8. e) A version of a stochastic process and a modification of a stochastic process. ANS: Detail Definitions 3.1.8 and 3.1.9. 2. (4+2+2+2+2+4) Let = (0 , 1] with the corresponding Borel field B and the (uniform) probability measure U on this measurable space....
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