Question

A communication link transmits binary characters, (0, 1). There is a probability p that a

transmitted character will be received correctly by a receiver, which then transmits to another
link, etc. If X0 is the initial character, and X1 is the character received after the first transmission,
X2 after the second, etc., then ( Xn ) is a Markov chain. Find the one step transition matrix and
the steady state probabilities.

Answer 1

The one-step transition matrix for the Markov chain (Xn) can be found by considering the probabilities of transitioning from one state to another. To find the steady state probabilities, we need to solve the equation π = π * P, where π is a row vector representing the steady state probabilities and P is the one-step transition matrix. This equation is solved by finding the eigenvector corresponding to the eigenvalue 1 of the transpose of the one-step transition matrix.

Step-by-step explanation:

The one-step transition matrix for the Markov chain (Xn) can be found by considering the probabilities of transitioning from one state to another. Since there are two possible states (0 and 1), the transition matrix will be a 2x2 matrix. Let's denote the transition probability from state 0 to state 0 as p00 and the transition probability from state 0 to state 1 as p01. Similarly, the transition probability from state 1 to state 0 is p10 and from state 1 to state 1 is p11. Thus, the one-step transition matrix will be:

p00

p01

p10

p11

To find the steady state probabilities, we need to solve the equation:

π = π * P

Where π is a row vector representing the steady state probabilities and P is the one-step transition matrix. Essentially, we want to find a vector π that when multiplied by P, gives us the same vector π. This equation is solved by finding the eigenvector corresponding to the eigenvalue 1 of the transpose of the one-step transition matrix. The steady state probabilities will be the elements of this eigenvector, normalized to sum to 1.