Question

Problem 4

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 0.99 that the digit entered will be transmitted accurately. In other words, the probability is 0.01 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) Define the variable and states for this system.

(b) Construct the (one-step) transition matrix.

(c) 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 transmisión.

Answer 1

Solution:

The transaction matrix is

square value of the above matrix is

From the question we required to calculate p¹⁰ for 10-step transition matrix P(10), we get data from logarithms

From the resultant p¹⁰ matrix we getting the probability of the correct result is

= p₀₀ or p₁₁ = 95.22%

Let assume that to getting more accurate result we replace 0.995 with 0.998 in above 'p' matrix the result as follows

From the question we required to calculate p¹⁰ for 10-step transition matrix P(10), we get data from logarithms

From the resultant p¹⁰ matrix we getting the probability of the correct result is

= p₀₀ or p₁₁ = 98.04%