(a) A two state Markov process may emit '0' in State 0 or emit '1' in State 1, each with probability a, and return to the same state or with probability 1 a it emits the other symbol and switches to the other state Thus it tends to be sticky or oscillatory, two forms of predictability, depending on a '0' 1 What are the state occupancy probabilities for 0 a 1 Two State Markov Process State 0 p 1 a '0' '1' p 1 a State 1 '1' 2 What are the entropy of State 0, the entropy of State 1, and the overall entropy of this source Express your answers in terms of a 3 For what value(s) of a do both forms of predictability disappear What then is the entropy of this source, in bits per emitted bit (b) Consider an alphabet of 8 symbols whose probabilities are as follows ABCDE F G H 128 1 If someone has selected one of these symbols and you need to discover which symbol it is by asking 'yes no questions that will be truthfully answered, what would be the most efficient sequence of such questions that you could ask in order to discover the selected symbol 2 By what principle can you claim that each of your proposed questions is maximally informative 3 On average, how many such questions will need to be asked before the selected symbol is discovered (c) Huffman trees enable construction of uniquely decodable prefix codes with optimal codeword lengths The five codewords shown here for the alphabet A,B,C,D,E form an instantaneous prefix code 1 Give a probability distribution for the five letters that would result in such a tree 2 Calculate the entropy of that distribution 3 Compute the average codeword length for encoding this alphabet Relate your results to the Source Coding Theorem A 00 B 01 C 10 D 110 E 111

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Question: (a) A two-state Markov process may emit '0' in State 0 or emit '1' in State 1, each with probability a, and return to

(a) A two-state Markov process may emit '0' in State 0 or

emit '1' in State 1, each with probability a, and return to other symbol and switches to the other state. Thus it tends to 1. What are the state occupancy probabilities for 0 < a

(a) A two-state Markov process may emit '0' in State 0 or emit '1' in State 1, each with probability a, and return to the same state; or with probability 1 - a it emits the other symbol and switches to the other state. Thus it tends to be "sticky" or oscillatory, two forms of predictability, depending on a. '0' 1. What are the state occupancy probabilities for 0 < a

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