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midterm1_S08

# midterm1_S08 - e[6 pt Problem 2 1 If X is a random variable...

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March 5, 2008 Spring 2008 Optimal Estimation (EGM 6934, Sec 4159 ) University of Florida Mechanical and Aerospace Engineering Instructor: Prabir Barooah Midterm 1 Duration: 50 minutes There are three problems that are worth 26, 14, and 20 points, respectively. Points will be awarded for clarity and completeness of your answers. Problem 1. The amount of time, in months, that a spacecraft functions before breaking down is modeled as an exponentially distributed continuous-type random variable X , whose p.d.f. is given by f X ( x ) = ( λe - λx x 0 0 x < 0 1. What is the Cumulative Distribution Function (CDF) of X ? [10pt] 2. Sketch the CDF. [10pt] 3. If λ = 1 12 . 2 , what is the probability that the spacecraft will function less than 12 . 2 months before breaking down? (you can leave your answer in terms of
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Unformatted text preview: e ) [6 pt] Problem 2. 1. If X is a random variable with variance σ 2 and a is a deterministic scalar, determine the variance of aX . [6 pt] 2. If X is a random vector with covariance matrix Σ and A is a deterministic matrix of appropriate dimension, determine the covariance matrix of A X . [8 pt] Problem 3. Let X be a random variable whose pdf is given by f X ( x | θ ) = ( 2 θ 2 x < x < θ otherwise Suppose a single observation x o of X is given (assume x o > 0). 1. What is the likelihood function ‘ ( θ | x o )? [7 pt] 2. Provide a sketch of ‘ ( θ | x o ). [7 pt] 3. Find the max-likelihood estimator of the parameter θ given the single observation x o . [6 pt] Prabir Barooah 1...
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• Fall '08
• Staff
• Probability theory, probability density function, Cumulative distribution function, Prabir Barooah, Aerospace Engineering Instructor

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