## exam198

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## exam198

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- Pages:
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- School:
- Washington University in St. Louis
- Course:
- Ese 523 - Information Theory

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EE 553 Exam 1 Professor Joseph A O Sullivan March 27 1998 This is a closed book closed notes exam It is scheduled to last for 1 5 hours All students must do their own work and are bound by the SEAS honor code If you do not undertand a question clearly state the assumptions used in trying to nd a solution NAME Potentially useful stu X1 k 0 X1 k k 0 k k 1 1 j j 1 1 1 1 2 j j 1 1 2 1 20 points Suppose that a channel has binary inputs X and binary outputs Y Suppose that the transition probabilities are P Y 1jX 1 0 8 P Y 0jX 1 0 2 P Y 1jX 0 0 1 P Y 0jX 0 0 9 Find the capacity of this channel 1 3 4 2 20 points Suppose that X is a geometric random variable so the probability that X k is given by PX k 1 k k 0 5 where 0 1 a Find the entropy of X b Describe the typical set Make sure that you convince me that you understand what the typical set looks like for this problem c On average how many bits does it take to represent n i i d random variables X1 X2 X d Would it be a good idea to use a Hu man code for data compression for a random variable X Explain your answer n 3 15 points Let X be a nite set and suppose that three probability distributions p x q x and r x are de ned on X Let 0 1 We will look at a few relative entropies and entropies D1 D2 D3 D4 D5 H1 H2 H3 D pjjr D qjjr D p 1 D p 1 D q 1 H p H q H p 1 qjjr rjjr rjjr q 6 7 8 9 10 11 12 13 Determine whether each of the following statements is true false or whether there is insu cient information undetermined to decide the truth of the statement Do not guess as wrong answers will be penalized Statement D3 D1 1 D2 D4 D1 D5 D1 D4 D5 D1 D2 H3 H1 1 H2 H1 D1 True False Undetermined 2 4 15 points Suppose that X f0 1 2g and that p 0 1 1 16 1 32 p 1 1 16 and p 2 1 32 a Describe a Hu man code for the random variable X b Find a Shannon code for the random variable X c Compare the two codes from parts a and b d Now suppose you were asked to nd a data compression code for 20 i i d copies of X That is the code should be de ned for the

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