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, D, 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 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

we have P[A]=1/4 P[B]=1/3 P[C]=1/6 P[D]=1/8 P[E]=1/8

after coding we get

A -> 1 length 1

B -> 01 length 2

C -> 001 length 3

D -> 0001 length 4

E -> 0000 length 4

Y is the length of the binary codeword produced in this situation.

a) so Y can take the values 1,2,3 and 4

b) Y<=2 means Y can take the values 1 or 2

now Y can take the value 1 when A has occurred

and Y can take the value 2 when B has occurred.

hence the corresponding event is AUB.

so P[AUB]=P[A]+P[B]=1/4+1/3=7/12 [answer]

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