a We wish to form an estimate of X 3 of the form ˆ X 3 aX 2 b where a and b are

# A we wish to form an estimate of x 3 of the form ˆ x

• mahsanabs
• 5

This preview shows page 3 - 5 out of 5 pages.

a) We wish to form an estimate of X 3 of the form ˆ X 3 = aX 2 + b where a and b are some constants. Find the value of a and b that minimize the mean squared error (MSE) given X 2 = x 2 . What is the corresponding MSE value? b) Find the minimum mean square estimator of X 3 given X 1 = x 1 and X 2 = x 2 . Find the corresponding value of MSE and compare that with the value in Part (a). Solution2 a) ˆ X 3 = E[ X 3 ] + Cov ( X 3 ,X 2 ) V ar ( X 2 ) ( X 2 -E[ X 2 ]) = ρ ( x 2 - 1) Var( ˆ X 3 ) = Cov ( X 3 , X 2 ) - Cov ( X 3 ,X 2 ) V ar ( X 2 ) = 1 - ρ 2 b) Similarly ˆ X 3 = ( 0 ρ ) 1 ρ ρ 1 - 1 x 1 - 1 x 2 - 1 Var( ˆ X 3 ) = 1 - ( 0 ρ ) 1 ρ ρ 1 - 1 0 ρ = (1 - ρ 2 1 - ρ 2 ) 1 - ρ 2 Page 3 of 5

Subscribe to view the full document.

ELEC 533 (Behnaam Aazhang ): Midterm Examination 2017 Problem 2 Problem 3 Suppose { X t ; t R } is a Gauss-Markov process, meaning that X t is a zero-mean wide sense stationary (WSS) Gaussian process with autocorrelation R X ( t, s ) = e -| t - s | . 1. Show that if you sample X t at three different points of time t 1 , t 2 , t 3 , then E [ X t 3 | X t 1 , X t 2 ] depends only on X t 2 , and not on X t 1 . 2. Show that X t 1 , X t 2 , and X t 3 form a Markov chain; that is, X t 3 conditioned on X t 2 is independent of X t 1 . Solution3 a) We will assume that: t 1 < t 2 < t 3 R x ( t, s ) = e -| t - s | = E [ X t X s ] because the mean is zero. Therefore we can write the following: R x ( t 1 , t 2 ) = R x ( t 2 , t 1 ) = e t 1 - t 2 and similar we can write: R x ( t 2 , t 3 ) = e t 2 - t 3 , R x ( t 1 , t 3 ) = e t 1 - t 3 . X (1) = X t 1 X t 2 ∼ N 0 0 , Σ 11 Σ 11 = R x ( t 1 , t 1 ) R x ( t 1 , t 2 ) R x ( t 2 , t 1 ) R x ( t 2 , t 2 ) = 1 e t 1 - t 2 e t 1 - t 2 1 det(Σ 11 ) = 1 - e 2( t 1 - t 2 ) Σ - 1 11 = 1 e t 1 - t 2 e t 1 - t 2 1 × 1 1 - e 2( t 1 - t 2 ) X (2) = X t 3 ∼ N (0 , Σ 22 ) with Σ 22 = R x ( t 3 , t 3 ) = 1.
• Spring '14
• Aazhang,Behnaam
• Probability theory, random process Xt, Behnaam Aazhang, different points of time t1

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes