Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 22

MATH152 April 2013

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[hide] Question A 22

Consider a random walk with three possible states and where the random walker (possibly) changes from state to state once every day according to the transition matrix P. Answer the following questions using the martix P and some of its powers given below. ${\begin{aligned}P={\begin{bmatrix}0.0284&0.5938&0.1626\\0.9207&0.1579&0.1634\\0.0509&0.2483&0.6740\end{bmatrix}}\\P^{10}={\begin{bmatrix}0.2832&0.2907&0.2875\\0.3843&0.3753&0.3786\\0.3325&0.3340&0.3339\end{bmatrix}}\\P^{20}={\begin{bmatrix}0.2874&0.2875&0.2875\\0.3791&0.3790&0.3790\\0.3335&0.3335&0.3335\end{bmatrix}}\end{aligned}}$ What is the probability that a walker starting in location 3 is in location 1 after one time step?

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[show] Hint
The probability of moving from location $\displaystyle i$ to location $\displaystyle j$ in one time step is $\displaystyle P_{j,i}$. This is the same as the i-th entry of the vector Pej.

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[show] Solution
The probability of moving from location 3 to location 1 in one time step is $\displaystyle P_{1,3}=0.1626$.

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