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# HW4 - EML 6934 Fall 2009 Optimal Estimation University of...

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EML 6934 Fall 2009 Optimal Estimation University of Florida Mechanical and Aerospace Engineering HW 4 Issued: September 18, 2009, Due: September 25, 2009 (in class) Note: Your work for the questions with “[0 pt]” are not to be turned in. Problem 1. [5 pt] Show that var ( aX ) = a 2 var ( X ) ( a is a deterministic parameter). Problem 2. [10 + 10 = 20 pt] Prove the CBS (Cauchy-Bunyakovskii-Schwarz) Inequality for random variables. That is, show that for two random variables X and Y , Covar ( X, Y ) radicalbig V ar ( X ) V ar ( Y ) and similarly | E[ XY ] | 2 E[ X 2 ] E[ Y 2 ] Problem 3. [10 + 10 = 20 pt] Show the following triangle inequality for random variables, that for two random variables X and Y , radicalbig E[( X + Y ) 2 ] radicalbig E[ X 2 ] + radicalbig E[ Y 2 ] and radicalbig V ar ( X + Y ) radicalbig V ar ( X ) + radicalbig V ar ( Y ) Problem 4 (Sample variance as an estimate of the variance) . [5+10+0 = 15 pt] Let X 1 , . . . , X n be n independent random variables that have the same mean μ = E[ X i ] and variance σ 2 = E[( X - μ ) 2 ], and μ < , σ 2 < . Now consider the sample mean ˆ μ and the sample variance ˆ σ 2 : ˆ μ = X 1 + X 2 + . . . X N N , ˆ σ 2 = ( X 1 - ˆ μ ) 2 + ( X 2 - ˆ μ ) 2 + · · · + ( X N - ˆ μ ) 2 N - 1 , 1. What is the variance of the sample mean?

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HW4 - EML 6934 Fall 2009 Optimal Estimation University of...

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