Question: 3.1.2 Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage

3.1.2 Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a fixed probability of error . Suppose that X0 D 0 is the signal that is sent and let Xn be the signal that is received at the nth stage. Assume that fXng is a Markov chain with transition probabilities P00 D P11 D 1???? and P01 D P10 D , where 0 < < 1.

(a) Determine PrfX0 D 0;X1 D 0;X2 D 0g, the probability that no error occurs up to stage n D 2.

(b) Determine the probability that a correct signal is received at stage 2.

Hint: This is PrfX0 D 0;X1 D 0;X2 D 0gCPrfX0 D 0;X1 D 1;X2 D 0g.

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