zz-13 - Y which is a whitened version of X . Problems from...

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Utah State University ECE 6010 Stochastic Processes Homework # 4 Due Friday Oct. 7, 2005 1. Suppose X ∼ N ( μ , Σ). (a) Show that E [ X ] = μ and cov( X , X ) = Σ. (b) Show that A X + b ∼ N ( A μ + b , A Σ A T ). (c) Suppose Σ > 0 and write Σ = CC T . Show that C - 1 ( X - μ ) ∼ N ( 0 , I ). 2. Suppose you have a random number generator which is capable of generating random numbers distributed as X ∼ N (0 , 1). Describe how to generate random vectors Y N ( μ , Σ). 3. Suppose X and Y are r.v.s. Show that E [( X - h ( Y )) 2 ] is minimized over all functions h when h is the function h ( y ) = E [ X | Y = y ] . Assume E [ X 2 ] < . 4. Let ( X, Y ) ∼ N ( μ x , μ y , σ 2 x , σ 2 y , ρ ). Let X = [ X Y ] ∼ N ( μ , Σ), where Σ = ± s 11 s 12 s 21 s 22 ² . Determine the relationship between μ x , μ y , σ 2 x , σ 2 y , ρ and μ and Σ. 5. Suppose X = h X 1 X 2 X 3 i ∼ N ( μ , Σ) where μ = 1 2 3 Σ = 4 2 1 2 6 3 1 3 8 (a) The value X 1 = 1 . 5 is measured. Determine the best estimate for ( X 2 , X 3 ). (b) In a separate problem, the values X 2 = 1 and X 3 = 5 are measured. Determine the best estimate of X 1 . 1
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(c) Determine a random vector
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Unformatted text preview: Y which is a whitened version of X . Problems from Grimmet & Stirzaker 1. Ex 3.7.5. What is requested is E [ T-t | T > t ], i.e., the mean subsequent lifetime given that the machine is still running after t days. Then use the hint from the book. Note that in (a), P ( T > t ) = 1 N +1 ( N-t ). 2. Ex 3.7.7. Hint: Show that P (robot faulty | fault not detected) = (1- ) 1- 4 = . Hence argue that the number of faulty passed robots, given Y , is distributed as B ( n-Y, ), which has mean ( n-Y ) . Hence show that E [ X | Y ] = Y + ( n-Y ) . 3. Ex 4.1.1(a) 4. Ex 4.1.2 5. Ex 4.2.1. Hint: think geometric r.v. 6. Ex 4.2.2. Hint: P (max( X, Y ) v ) = P ( X v, Y v ). 7. Ex 4.4.1. Hint: integrate by parts. 8. Ex 4.6.4(b) 2...
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This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.

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zz-13 - Y which is a whitened version of X . Problems from...

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