1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. • If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode. If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking X, to be the state of the machine on the morning of day i for i E N we get a Markov chain which models the state of the machine.
1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. • If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode. If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking X, to be the state of the machine on the morning of day i for i E N we get a Markov chain which models the state of the machine.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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(c) Calculate the expectation of the number of days of smooth running in the
lifetime of the machine.
(d) The following mathematical expression describes an event. Give a description
of this event as you would express it to a non-mathematician:
|{n € {0, 1,. ‚T − 1} : X₂ = 1}| > |{n € {0, 1, ...,T – 1} : X₂ = 2}|
where T min{n: Xn = 4}.
..."
Transcribed Image Text:1.
A machine can be in one of four states: 'running smoothly' (state
1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and
'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose
that the state of the machine tomorrow morning depends only on the state of the
machine this morning subject to the following rules.
• If the machine is running smoothly, there is 1% chance that by the next morning
it will have exploded (this will destroy the machine), there is also a 9% chance
that some part of the machine will break leading to it being temporarily broken.
If neither of these things happen then the next morning there is an equal
probability of it running smoothly or running but needing adjustment.
• If the machine is temporarily broken in the morning then an engineer will
attempt to repair the machine that day, there is an equal chance that they
succeed and the machine is running smoothly by the next day or they fail and
cause the machine to explode.
. If the machine is running but needing adjustment there is a 20% chance that
an engineer will repair it so it is running smoothly the next day and otherwise
it will remain in the same state for the next day.
Taking X₂ to be the state of the machine on the morning of day i for i E N we get a
Markov chain which models the state of the machine.
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