ps7 - ECE 804 Random Signal Analysis OSU Autumn 2008 Nov 10...

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ECE 804, Random Signal Analysis Nov. 10, 2008 OSU, Autumn 2008 Due: Nov. 17, 2008 Problem Set 7 Problem 1 We want to obtain the mold content per volume, m , of the water in the Dreese building, with an error that, with 95 % probability, is less than 0 . 1. The technique we use for this measurement has an error that is random with mean 0 and standard deviation 2. So, we can model our measurements as X i = m + N i , where N i is the noise in measurement i , with a mean of 0, and a std. deviation of 2. The N i ’s are independent for i = 1 , 2 , . . . . In order to reduce the error, we perform a number of measurements and compute their average: M n = 1 n n X i =1 X i (a) Find the mean and the variance of M n . (b) Suppose we model M n as a Gaussian random variable. With this approximation, what is the number of measurements needed to achieve the desired reliability? (c) Using the Chebychev Inequality, find an upper bound on the number of measurements we need to achieve the reliability goal. Problem 2 Consider the function (for c > 0) f ( x, y ) = ( 2 - | x | , - c x c, - 1 y 1 0 , otherwise (a) Find the constant c for which f ( x, y ) is the joint pdf of two random variables.
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