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Markov Chains
From weather conditions to baseball scores to stock performances, many probabilistic real-world systems can be modeled with Markov chains.
A Markov chain has transition matrix \[\begin{pmatrix} 0.5 & 0.3 & 0.2 \\ 0.4 & 0.4 & 0.2 \\ 0.6 & 0.4 & 0 \end{pmatrix}.\]
What is its \(2\)-step transition matrix?
(A): \(\begin{pmatrix} 0.49 & 0.35 & 0.16 \\ 0.48 & 0.36 & 0.16 \\ 0.46 & 0.34 & 0.2 \end{pmatrix}\)
(B): \(\begin{pmatrix} 0.25 & 0.35 & 0.4 \\ 0.39 & 0.25 & 0.36 \\ 0.36 & 0.39 & 0.25 \end{pmatrix}\)
(C): \(\begin{pmatrix} 0.49 & 0.35 & 0.16 \\ 0.48 & 0.36 & 0.16 \\ 0.48 & 0.32 & 0.2 \end{pmatrix}\)
(D): \(\begin{pmatrix} 0.49 & 0.35 & 0.16 \\ 0.4 & 0.44 & 0.16 \\ 0.48 & 0.32 & 0.2 \end{pmatrix}\)
Phineas is flipping a fair coin repeatedly and marking down the results. However, he has a soft spot for heads, so half the time he marks heads, he marks heads again. (This means he could possibly mark down heads many times in a row without actually flipping the coin again.) The resulting sequence of states could be modeled by a Markov chain, where the first possible state is heads and the second possible state is tails. What is its transition matrix?
(A): \(\begin{pmatrix} \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} \end{pmatrix}\)
(B): \(\begin{pmatrix} \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{4} & \tfrac{3}{4} \end{pmatrix}\)
(C): \(\begin{pmatrix} \tfrac{3}{4} & \tfrac{1}{2} \\ \tfrac{1}{4} & \tfrac{1}{2} \end{pmatrix}\)
(D): \(\begin{pmatrix} \tfrac{3}{4} & \tfrac{1}{4} \\ \tfrac{1}{2} & \tfrac{1}{2} \end{pmatrix}\)
A Markov chain can be constructed for the weather based on the following table.
| States | Rainy Tomorrow | Cloudy Tomorrow | Sunny Tomorrow |
| Rainy Today | \(\tfrac{1}{2}\) | \(\tfrac{1}{3}\) | \(\tfrac{1}{6}\) |
| Cloudy Today | \(\tfrac{1}{3}\) | \(\tfrac{1}{3}\) | \(\tfrac{1}{3}\) |
| Sunny Today | \(0\) | \(\tfrac{1}{9}\) | \(\tfrac{8}{9}\) |
If it is rainy today, what is the probability that it will be rainy in three days?
If a Markov chain \(\{X_0, \, X_1, \, \dots\}\) has transition matrix \(P\) and states \(i\) and \(j\), then what is the value of \(P_{i,j}\)?