⧼exchistory⧽
BBy Bot
Jun 09'24
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In Example, for what values of [math]a[/math] and [math]b[/math] do we obtain an absorbing Markov chain?

BBy Bot
Jun 09'24
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Show that Example is an absorbing Markov chain.

BBy Bot
Jun 09'24
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Which of the genetics examples (Example and Example) are absorbing?

BBy Bot
Jun 09'24
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Find the fundamental matrix [math]\mat{N}[/math] for Example.

BBy Bot
Jun 09'24
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For Example, verify that the following matrix is the inverse of [math]\mat{I} - \mat{Q}[/math] and hence is the fundamental matrix [math]\mat{N}[/math].

[[math]] \mat{N} = \pmatrix{ 8/3 & 1/6 & 4/3 & 2/3 \cr 4/3 & 4/3 & 8/3 & 4/3 \cr 4/3 & 1/3 & 8/3 & 4/3 \cr 2/3 & 1/6 & 4/3 & 8/3 \cr}\ . [[/math]]

Find [math]\mat{N} \mat{c}[/math] and [math]\mat{N} \mat{R}[/math]. Interpret the results.

BBy Bot
Jun 09'24
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In the Land of Oz example (Example), change the transition matrix by making R an absorbing state. This gives

[[math]] \mat{P} = \bordermatrix{ & \mbox{R} & \mbox{N} & \mbox{S} \cr \mbox{R} & 1 & 0 & 0 \cr \mbox{N} & 1/2 & 0 & 1/2 \cr \mbox{S} & 1/4 & 1/4 & 1/2}\ . [[/math]]

Find the fundamental matrix [math]\mat{N}[/math], and also [math]\mat{Nc}[/math] and [math]\mat{NR}[/math]. Interpret the results.

BBy Bot
Jun 09'24
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In Example, make states 0 and 4 into absorbing states. Find the fundamental matrix [math]\mat{N}[/math], and also [math]\mat{Nc}[/math] and [math]\mat{NR}[/math], for the resulting absorbing chain. Interpret the results.

BBy Bot
Jun 09'24
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In Example (Drunkard's Walk) of this section, assume that the probability of a step to the right is 2/3, and a step to the left is 1/3. Find [math]\mat{N},~\mat{N}\mat{c}[/math], and [math]\mat{N}\mat{R}[/math]. Compare these with the results of Example.

BBy Bot
Jun 09'24
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A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State five is an absorbing state. Find the expected number of steps to reach state five.

BBy Bot
Jun 09'24
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Using the result of Exercise, make a conjecture for the form of the fundamental matrix if the process moves as in that exercise, except that it now moves on the integers from 1 to [math]n[/math]. Test your conjecture for several different values of [math]n[/math]. Can you conjecture an estimate for the expected number of steps to reach state [math]n[/math], for large [math]n[/math]? (See Exercise for a method of determining this expected number of steps.)