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# hw4 - CS112 Homework#4 1 The per bit error rate over a...

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CS112 - Homework #4 1. The per bit error rate over a certain binary communication channel is 10 - 10 . No other statistics are known about the channel or the data. (a) What is the expected number of erroneous bits in a block of 1000 bits? (Ans: 10 - 7 ) (b) Find an upper bound on the probability that a block of 1000 bits has 10 or more erroneous bits.Hint: Use Markov Inequality. 2. Consider a CD-ROM disk with a radius r .Typically in CD-ROM drives, there is a laser disk reader that reads the “pits” on the disk. Assume that the laser reader always beams on the disk, and that all the points of impact ( x, y ) are equally likely, so that the joint PDF of the random variables X and Y is uniform. (a) Find f X,Y ( x, y ) ? (b) Find f Y ( y ) . Is it a uniform PDF? (c) Find f X | Y ( x | y ). Is it a uniform PDF? 3. A computer system consists of n subsystems, each of which has an exponential distribution of lifetime with parameter λ i , for i = 1 , 2 , . . . , n . Each subsystem is independent but the whole computer system fails if any of the subsystems fails. Show that the lifetime of the computer system follows an exponential random variable with parameter n i =1 λ i . 4. There are n hard disks. Let X 1 , . . . , X n be independent random variables representing the lifetime (in years) of the disks. Each X i has a uniform distribution over (0 , 1) for i = 1 . . . n . Let M = max { X 1 , . . . , X n } , the random variable representing the lifetime of the hard disk that fails the last. Show that the distribution function of M is given by F M ( x ) = x n , 0 x 1.

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