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Unformatted text preview: ACM116  Winter 20082009  Homework #4 Handed out: March 6, 2009, Due: March 18, 2009, in Sheila Shull’s office (Firestone 217) by 4pm Please write down your solutions clearly and concisely, put the problems in order, and box your answers. For questions regarding each problem, please contact the TA whose name is printed in front of the problem index. To get full credit it is sufficient to solve Problem 1 to 7. Solving Problem 8 is optional and will give you extra points. 1/(Yaniv) Suppose X ∼ N ( μ, Σ). Let A be an n × n matrix with the property AA T = Σ. Prove that the random vector Z = A 1 ( X μ ) is a standard normal random vector. 2/(Yaniv) Suppose X ∼ N (0 , Σ), with Σ = 4 1 1 4 (a) What is the law of X 1 + X 2 ? (b) Write X as the sum of two independent Gaussian vectors. (c) What is E [ X 1 X 2  X 1 + X 2 ]? (d) What is the Law of ( X 1 X 2  X 1 + X 2 )? 3/(Yaniv) Suppose X is a signal of interest, that has modeled as a Gaussian vector: X ∼ N (0 , Σ) with Σ = 3 1 1 1 3 1 1 1 3 Suppose the signal is corrupted with noise, so you see Y = X + Z , where Z is a standard normal noise vector ( Z ∼ N (0 ,I )). If, in one instance, Y = (3 , 1 , 2), what is your best approximation of X ? Hint: Use the Wiener filter. 1 4/(Stephen) Let B be a Brownian Motion. Consider the stochastic process X t defined by for t ∈ T with T = [0 , 1] by X t = B t tB 1 a Show that ( X t ) t ∈ T is a centered Gaussian Process. b Compute its covariance matrix Γ[ s,t ] = E [ X s X t ]. c Is the law of stochastic process X t equal to the law of a Brownian Motion on [0 , 1]? d Show that ( X t ) t ∈ T is independent from B 1 . e Let φ be a bounded measurable function from R n onto R + . Write F [ x ] := E [ φ ( X t 1 + t 1 x,...,X t n + t n x )] What is the value of E [ φ ( X t 1 + t 1 B 1 ,...,X t n + t n B 1 )  B 1 ]?...
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 Spring '10
 323
 Normal Distribution, Brownian Motion, Probability theory, WI

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