Question

Suppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10 times in succession. During each transmission, the probability is 0.995 that the digit entered will be 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 Xo 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, and so on, then {XnJ is a Markov chain. (a) Construct the (one-step) transition matrix (b) Find the 10-step transition matrix. 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 trans- mission will be accurate from 0.995 to 0.998. Repeat part (b) to find the new prob- ability that a digit entering the network will be recorded accurately after the last transmission

Answer 1

A.) One step transition matrix:

We know p(1|1) =p(0|0) = 0.995. So,

P(1|0) = p(0|1) = 1-0.995 = 0.005

Then the transition matrix is

Received sent 0 1

0 0.995 0.005

1 0.005 0.995

Similarly we can expand the matrix for ten step transition matrix.