Chapter 5 Probability Models

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Chapter 5 Probability Models

• Introduction
• Modeling situations that involve an element of
  chance
• Either independent or state variables is
  probability or random variable
• Markov chain
• Random variables
• Statistics, system reliability,
• Game theory
• Casino,

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An example

• The Problem --- Consider a car rental company
  with branches in Orlando and Tampa. Each branch
  rents car to Florida tourists. The company
  specializes in catering to travel agents who want
  to arrange tourist activities in both Orlando and
  Tampa. Consequently, a traveler will rent a car
  in one city and drop off the car in either city.
  Travelers begin their itinerary in either city.
  Cars can be returned to either location, which
  can cause an imbalance in available cars to rent.
  Historical data on the percentages of cars rented
  and returned to these companies is collected from
  the previous few years as shown here.

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An example transition matrix
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An example transition matrix

• The recorded data form a transition matrix and
  show that the probability for
• Returning a car to Orlando that was rented in
  Orlando is 0.6 whereas the probability it will
  be returned in Tampa is 0.4.
• Returning a car to Orlando that was rented in
  Tampa is 0.3 whereas the probability it will be
  returned in Tampa is 0.7.
• There are two states Orlando and Tampa
• The sum of the probabilities for transitioning
  from a present state to next state, which is the
  sum of the probabilities in each row, 1--all
  possible outcomes are taken into account.

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An example -- model

• Variables
• pn --- the percentage of cars available to rent
  in Orlando at the end of period n
• qn --- the percentage of cars available to rent
  in Tampa at the end of period n
• Probabilistic model

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An example model solution
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An example model solution
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An example model interpretation

• If the two branches begin the year with a total
  of n cars, after 14 time periods or days
  approximately 57 of cars will be in Tampa and
  43 will be in Orlando.
• Starting with 100 cars in each location, about
  114 cars will be based out of Tampa and 86 will
  be based out of Orlando in the steady state (and
  it would take about 5 days to reach this state)

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Markov chain

• A Markov chain is a process in which there are
  the same finite number of states or outcomes that
  can be occupied at any given time. The states do
  not overlap and cover all possible outcomes.
• In a Markov process, the system may move from one
  state to another one at each time step, and there
  is a probability associated with this transition
  for each possible outcome. The sum of the
  probabilities for transitioning from the present
  state to the next state is equal to 1 for each
  state at each time step.

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A Markov process with two states
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Markov chain

• A sequence of events with the following
  properties
• An event has a finite number of outcomes, called
  states. The process is always in one of these
  states
• At each stage or period of the process, a
  particular outcome can transition from its
  present state to any other state or remain in the
  same state.
• The probability of going from one state to
  another in a single state is represented by a
  transition matrix for which the entries in each
  row lie between 0 and 1 each row sums to 1.
  These probabilities depend only on the present
  state and not on past states.

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Application of Markov chain

• In biology
• Color of a plant root --- yellow or green
• Color of pig hair --- black or white
• These can be explained by Markov chain
• The outside character of a species is determined
  by its genome which can be divided into two kinds
  dominate (d) and reminder (r). For each outside
  character, there is two genomes and each can be
  either d or r.

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Application of Markov chain

• Three combinations
• d d --- good --- denoted by D
• d r --- mixture -- denoted by H
• r r --- worse --- denoted by R
• From the biology theory
• With either d d or d r genome combination, the
  outside character shows dominate (or d), e.g.
  green in plant.
• With r r genome combination, the outside
  character shows reminder (or r), e.g. yellow in
  plant.

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Application of Markov chain

• In biology theory
• A child randomly picks up one genome from the two
  genomes of his father and one genome from the two
  genomes of his mother to form its own genome.
• Three states D, H and R
• Case 1 Parents are both D, child is D
• Case 2 Parents are DH, child is either D or H
  with probability ½ and ½
• Case 3 Parents are HH, child is D or H or R
  with probability ¼, ½ and ¼

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Application of Markov chain

• Problem 1 Marriage with mixture state
• Random pick up one, marry it with a mixture H,
  for their children, marry them again with
  mixture, and continue. What is probability of
  children shows outside characters d and r?
• Solution There are three states D, H R
• For each marry, the transition matrix is

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Application of Markov chain

• Model
• Limiting behavior
• Conclusion after many generations
• The probability of children shows outside
  character d is 0.250.5 0.75, and outside
  character r is 0.25.
• This theoretical results agree well with
  experiments!!

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Application of Markov chain

• Problem 2 Marriage with relatives
• The parents at the beginning can be either good,
  mixture or worse, they have many children. The
  marriage is only taken between children and
  continue. What happens after many generations?
• States of parents --- D, H or R
• States of children DD, RR, DH, DR, HH, HR

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Transition matrix
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Application

• Model
• Limiting behavior, when n is very large
• Conclusion After several generations, all
  children will be either only good or only worse
  and will keep it forever!! No biodiversity!!

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System reliability

• Recent Toyota call home problem
• Computer, automobile, etc. are complicated
  system! If they perform well for a reasonably
  long period of time, we say these system are
  reliable!!
• The reliability of a component or system is the
  probability that it will not fail over a specific
  time period.
• Define f(t) to be the failure rate of an item,
  component, or system over time t, thus f(t) is a
  probability distribution.

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System reliability

• Let F(t) be the cumulative distribution function
  corresponding to f(t).
• Define the reliability of the item, component, or
  system by
• For a complicated system, if we know the
  reliability of each component, then we can build
  simple models to examine the reliability of
  complex systems.

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System reliability

• A complicated system consists of many small parts
  via series, parallel or combinations
• Connected by series A series system is
  consisted of several independent components and
  it becomes failure due to the failure of any one
  of the independent component
• An example A NASA space shuttles rocket
  propulsion system

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System reliability

• System reliability
• Connected by parallel it is a system that
  performs as long as a single one of its
  components remains operational.
• An example

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System reliability

• System reliability
• Connected by combination combining series and
  parallel relationships

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System reliability

• Exercise find the system reliability