Suppose that computer A sends each message to computer B simultaneously over two

Question

Suppose that computer A sends each message to computer B simultaneously over two unreliable telephone lines. Computer B can detect when errors have occurred in either line Let the probability of message transmission error in line 1 and line 2 be q1 and q2 respectively. Computer B requests retransmissions until it receives an error-free message on either line (a) Find the probability that more than k transmissions are required (b) Find the probability that in the last transmission, the message on line 2 is received free of errors

Explanation / Answer

P (j transmissions) = (q1q2) j1 (1 q1q2)

For geometric probabilities, we have the general result that

P (more than k transmissions) = (q1q2) k

P (line 2 error-free|m transmissions) = P (line 2 error-free and m transmissions) /P (m transmissions)

= (q1q2) m1 (1 q2) / (q1q2) m1 (1 q1q2)

= (1 q2) / (1 q1q2)

An information source produces symbols at random from a 5-letter alphabet: S = {a, b, c, d, e}.

The probabilities of the symbols are p (a) = 1 /2, p (b) = 1 /4, p(c) = 1 /8, p (d) = p (e) = 1 /16

A data compression system encodes the letters into binary strings as follows:

a 1, b 01 ,c 001, d 0001, e 0000

Let the random variable Y be equal to the length of the binary string output by the system. Specify the sample space of Y , SY , and the probabilities of its values. Clearly, Y can take values from the discrete set

SY = {1, 2, 3, 4}.

P(Y = 1) = P({a}) = 1/ 2

P(Y = 2) = P({b}) = 1/ 4

P(Y = 3) = P({c}) = 1 /8

P(Y = 4) = P({d, e}) = 2/ 16 = 1 /8

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