Digital communication system, Assignments of Digital Communication Systems

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P ~The noise voltage in an electric circuit can be modeled as 2 Gaussian random variable with mean equal to zero and variance equal io 1078. 1, What is the probability that the value of the noise exceeds 10 4? What is the probability that it exceeds 4 x 10-4? What is the probability that the = noise value is between - 2 x 10~* and 10747 7 2, Given that the valuc of the noise is positive, what is the probability that it exceeds J 0-47 3. This noise passes through a hal{-wave rectifier with characteristics x, x>0 ac = {% x <0 Find the PDF of the rectified noise by first finding its CDF. Why can we not use the general expression in Equation (4.1.10) here? 4, Find the expected value of the rectified noise in the previous part. 5. Now assume that the noise passes through a full-wave rectifier defined by g(x) = la}. Find the density fumction of the rectified noise in this case. What is the expected value of the output noise in this case? P 2- Let ¥ be a positive valued random variable; Le, fp(y) = Ofory <0. 1. Let w be any positive constant, Show that P(Y >a) < AE (Markoy inequality). . 2. LetX be any random variable with variance 0? and define Y = (X — E[X yp and.@ = ¢* for some «. Obviously the conditions of the problem are satistied for ¥ and o as chosen here. Derive the Chebychev inequality 2 PIX- EX >< % e2 FP 3