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

Question

A certain data source produces messages that can be one of five possibilities selected from the set {A, B, C, D, E}. The random experiment consists of observing the value that the data source actually produces. The probability of the five possibilities being generated is P ({A}) = 1/4, P ({B}) = 1/3, P({C}) = 1/6 and P({D}) = P ({E}) = 1/8. 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 A rightarrow 1 B rightarrow 01 C rightarrow 001 D rightarrow 0001 E rightarrow 0000. A random variable Y is defined that corresponds to the length of the binary codeword produced in this situation. What possible values can Y be? What is the event Y lessthanorequalto 2, and what is the probability of this event ?

Explanation / Answer

as code word for A has lenght 1, for B it is 2, for C , it is 3 , for D it is 4 and for E it is 4

a) hence Y as a variable can take values from 1to 4.

b|) Y<=2 is the event in which length of codeword can be at most 2.

P(Y<=2) =P(probabilty that Y has length 0+probabilty that Y has length 1+probabilty that Y has length 2)

=(1/4)+(1/3) =(4+3)/12 =7/12

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