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HW3 - 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 3 Issued: September 11, 2009, Due: September 18, 2009 (in class) Problem 1. [5 pt] If X N ( μ, σ ), compute P ( | X μ | > 3 σ ). If you need tabulated values of erf ( x ), you can find them in the reading material available in the e-learning website (chapter 2) Problem 2. [5+5 = 10 pt] Show that if Y N ( μ, σ ), then E[ Y ] = μ and var ( Y ) = σ 2 . Problem 3. [10 pt] If X is a random variable that is uniformly distributed between a and b , and γ is a real constant, show that γX is uniformly distributed between γa and γb . (hint: define Z Δ = γX . Then F Z ( z ) = P ( Z z ) = P ( X z/γ ) = F X ( . . . ) = .. ) Problem 4. [10 pt] Compute the mean and variance of a random variable X that is uniformly distributed between a and b . Problem 5. [5 pt] Show that for a r.v. X , V ar ( X ) = E[ X 2 ] ( ¯ X ) 2 . [5 pt] Problem 6. [5+5 = 10 pt] The unit square U in R 2 is the region such that the points in this region have their x - and y - coordinates between 0 and 1. Let X be random vector that is uniformly distributed in the unit square.
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