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# tut09 - You are told to assume that the quality of each...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Tutorial 9 April 20-21, 2006 1. Signal-to-Noise Ratio: If random variable X has mean µ n = 0 and standard deviation σ > 0, the ratio r = | µ | is called the measurement signal-to-noise ratio , or SNR , of X . The idea is that X can be expressed as X = µ + ( X µ ), with µ representing a deterministic, constant- valued “signal” and ( X µ ) the random, zero-mean “noise.” If we deFne | ( X µ ) | = D as the relative deviation of X from its mean µ , show that for α > 0, 1 P ( D α ) 1 . r 2 α 2 2. In your summer internship, you are working for the largest producer of lightbulbs. Your manager asks you to estimate the quality of the production; that is, to estimate p , the probability of a bulb produced by the factory to be defectless.
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Unformatted text preview: You are told to assume that the quality of each bulb is independent, and identically distributed. (a) Supposing you test n randomly picked bulbs, what is a good estimate for p , Z n , such that Z n converges to p in probability? (b) The management asks that the estimate is located in the range p ± . 1 with probability 0.95. Are 27 randomly picked bulbs enough for this speciFcation? Give the reason. 3. p X (x) n p Y n 1 1 1- -1- -n n 1 -n-1 n (y) 1 x n y Let X n and Y n have the distributions shown above. (a) Evaluate the expectation and variance for X n and Y n . (b) What does the Chebyshev inequality tell us about the convergence of X n ? (c) What does the Chebyshev inequality tell us about the convergence of Y n ? (d) Is X n convergent in probability? If so, to what value? Explain. (e) Is Y n convergent in probability? If so, to what value? Explain....
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