9. A certain data source produces messages that can be one of five possibilities

Question

9. A certain data source produces messages that can be one of five possibilities selected from the set {A, B, C, D, Ey. The random experiment consists of observing the value that the data source actually produces. The probability of the five possibilities being generated is 16 The message produced is to be transmitted over a binary communication system, and to do so it is necessary to represent every message using codeword composed of a string of binary values The code chosen is A1 B01 C001 D0001 E 0000. A random variable Y is defined that corresponds to the length of the binary codeword produced in this situation (a) The sample space for this experiment is S = {A,B,C,D,E). what is the mapping on S that is the random variable Y? (b) What is the event "Y

Explanation / Answer

Now P(A) = 1/2, P(B) = 1/4, P(C)=1/8, P(D) = P(E) = 1/16 where A,B,C,D and E are one of the five possibilities selected from the set {A,B,C,D,E}

Now the correspondence is A->1, B->01, C->001, D->0001, E->0000 which are the binary codes

a. Now S= {A,B,C,D,E}, So the mapping on S of the random variable Y = {1, 2,3,4,4} as these are the number of digits of the binary code that correspond to the sample space {A,B,C,D,E} of S. It is a direct one to one mapping

b. The event Y<=3, is equal to the events that either A, B or C occurs i.e. the datasource produces messages among these 3 possibilities, The probability of this event therefore is

P(A) +P(B) +P(C) = 0.5+0.25 + 0.125 = 0.875

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