Chapman-Kolmogorov EquationsSuppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10times in succession. During each transmission, the probability is 0.995 that the digit entered will betransmitted accurately. In other words, the probability is 0.005 that the digit being transmitted will berecorded with the opposite value at the end of the transmission. For each transmission after the first one, thedigit entered for transmission is the one that was recorded at the end of the preceding transmission. If X0denotes the binary digit entering the system, X1 the binary digit recorded after the first transmission, X2 thebinary digit recorded after the second transmission, . . . , then { Xn} is a Markov chain. A. Construct the (one-step) transition matrix. B. Find the 10-step transition matrix P(10). Use this result to identify the probability that a digit enteringthe network will be recorded accurately after the last transmission. C. Suppose that the network is redesigned to improve the probability that a single transmission will beaccurate from 0.995 to 0.998. Repeat part (b) to find the new probability that a digit entering the networkwill be recorded accurately after the last transmission.
Chapman-Kolmogorov Equations
Suppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10
times in succession. During each transmission, the
transmitted accurately. In other words, the probability is 0.005 that the digit being transmitted will be
recorded with the opposite value at the end of the transmission. For each transmission after the first one, the
digit entered for transmission is the one that was recorded at the end of the preceding transmission. If X0
denotes the binary digit entering the system, X1 the binary digit recorded after the first transmission, X2 the
binary digit recorded after the second transmission, . . . , then { Xn} is a Markov chain.
A. Construct the (one-step) transition matrix.
B. Find the 10-step transition matrix P(10). Use this result to identify the probability that a digit entering
the network will be recorded accurately after the last transmission.
C. Suppose that the network is redesigned to improve the probability that a single transmission will be
accurate from 0.995 to 0.998. Repeat part (b) to find the new probability that a digit entering the network
will be recorded accurately after the last transmission.
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