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Chapter 10 Markov Chains

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Then find the probabilities associated with a third and fourth purchase from the bookstores. ... those who bought from Campus Bookstore, 90% would buy from it ... – PowerPoint PPT presentation

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Title: Chapter 10 Markov Chains


1
Chapter 10Markov Chains
  • Section 10.1
  • Basic Properties
  • of Markov Chains

2
Some Background Information
  • Mathematical models that evolve over time in a
    probabilistic manner are called stochastic
    processes.
  • A special kind of stochastic process is a Markov
    Chain, where the outcome of an experiment depends
    only on the outcome of the previous experiment.

3
Why Study Markov Chains?
  • Markov chains are used to analyze trends and
    predict the future. (Weather, stock market,
    genetics, product success, etc.

4
A Sociology Example
Sociologists classify people by income as
lower-class, middle-class, and upper-class.
They have found that the strongest determinant
of the income class of an individual is the
income class of that persons parents.
5
Transition Diagrams
6
Transition Matrices
7
Characteristics of a Transition Matrix
  • It is a square matrix.
  • All entries are between 0 and 1 (because all the
    entries are probabilities).
  • The sum of the entries in any row must be 1.
  • Denoted by P .

8
Key Features of Markov Chains
  • A sequence of trials of an experiment is a Markov
    chain if
  • 1.) the outcome of each experiment
  • is one of a set of discrete states
  • 2.) the outcome of an experiment
  • depends only on the present state,
  • and not on any past states
  • 3.) the transition probabilities remain
  • constant from one transition to the
  • next.

9
Transition Probabilities from One State to
Another
  • The transition matrix shows the probabilities of
    moving from state-to-state from the current
    generation to the next.
  • P gives the probabilities of a transition from
    one state to another in k repetitions of an
    experiment, provided the transition probabilities
    remain constant from one repetition to the next.

k
10
Example 1
  • Use the transition matrix from the Sociology
    example to find the probabilities of change for
    the grandchildren of the current generation.

11
Example 2
  • Write a transition matrix for the following
    situation. Then find the probabilities
    associated with a third and fourth purchase from
    the bookstores.
  • In a study of the market share of the three
    bookstores in a university town, it was found
    that 75 of those who had bought from University
    Bookstore would buy from it again, 15 from
    Campus Bookstore and 10 from Bookmart.
  • Of those who bought from Campus Bookstore, 90
    would buy from it again and 5 each would buy
    from University Bookstore and Bookmart.
  • 85 of those who bought from Bookmart would buy
    from it again, 5 from University Bookstore and
    10 from Campus Bookstore.

12
Probability Vector (Matrix)
  • When a study is first begun, the probabilities of
    the states are called the initial distribution.
  • This distribution is written as a matrix of only
    one row.
  • Denoted by X0 .

13
Characteristics of aProbability Vector
  • It is a row matrix.
  • Each entry must be between 0 and 1 inclusive.
  • The sum of the entries of the row must be 1.

14
Making Predictions about the Population
Proportion with Markov Chains
  • 1.) Create a probability vector, X0 .
  • The entries are the initial probabilities of
    the states.
  • 2.) Create the transition matrix, P .
  • The entries are the probabilities of passing
    from current states (rows) to the first
    following states (columns).
  • 3.) Calculate X0 P
  • This is the probability vector after n
    repetitions of
  • the experiment. In other words, it is a
    prediction
  • for the proportion of the population after n
  • repetitions.
  • Note The columns and rows of the probability
    vector and the transition matrix must be labeled
    the same.

n
15
Example 3
  • A marketing analysis shows that KickKola
    currently commands 14 of the cola market.
  • Further analysis indicates that 12 of the
    consumers who do not currently drink KickKola
    will purchase KickKola the next time they buy a
    cola (in response to a new advertising campaign)
    and that 63 of the consumers who currently drink
    KickKola will purchase it the next time they buy
    a cola.
  • Predict KickKolas market share at
  • a.) the next following purchase
  • b.) the second following purchase

16
Example 4
  • It has been noted that George the golfer tends to
    repeat himself. In his first shot, he will hit
    his ball in the fairway (F), in the rough (R), or
    out of bounds (B).
  • On his first shot he hits in the fairway 60 of
    the time, in the rough 30 of the time, and out
    of bounds 10 of the time.
  • However, on subsequent shots, if he hit in the
    fairway the first time, he will hit in the
    fairway next time 90 of the time and in the
    rough 10 of the time.
  • If he hit in the rough the first time, he will
    hit the rough next time 50 of the time and out
    of bounds 5 of the time.
  • If he hits out of bounds the first time, he will
    hit out of bounds next time 70 of the time and
    in the fairway 15 of the time.
  • Find the probability that his next following
    shot is in
  • a.) the fairway
  • b.) the rough
  • c.) out of bounds
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