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I am new to stochastic processes and trying to solve a question related to finding a transition matrix of some experiment. The question is a

A sequence of experiments is performed, in each of which two fair coins are tossed. Let S1 indicate that two heads come up, S2 that a head and a tail come up. and S3 that two tails turn up. Find the transition matrix.

What I think is it is: $\left(\matrix{1/4 & 1/4 & 0 \\ 0 & 1/2 & 1/4 \\0 & 1/2 & 1/2}\right)$, but the sum of each row in transition matrix should be one. So my answer is not correct. Does anybody know how I can find these probabilities. Any help would be appreciated.

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1 Answer 1

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Given a stochastic process $\{X_n\}_{n \in \mathbb{N}}$ as described, see that each $X_i$ and $X_j$ are independent of each other, if $i \neq j$.

Our answer, then, should be

$$\begin{bmatrix} 1/4 & 1/2 & 1/4\\ 1/4 & 1/2 & 1/4\\ 1/4 & 1/2 & 1/4\end{bmatrix}$$

To see where you made a mistake, notice that in the first row of your answer, for example, you are saying that

$$P(X_{i+1} = S_3|X_i= S_1) = 0.$$

In other words, you are saying that if in the current outcome resulted in two heads, then it's impossible for the next outcome to result in two tails, which is clearly wrong.

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