Proposal
Number Sense
Interactive Quiz
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Glossary
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Andrei Andreyvich Markov,
commonly called a number theorist and mathematician, was born to his
parents, Andrei Grigorievich, an employee of the state forest
department, and Nadezhda,
who grew up in Russia. He
excelled in school, with the exception of mathematics.
After receiving his bachelor’s |
degree
at the University of St.
Petersburg, he continued his studies and earned his master’s degree
and Ph. D. During his
studies at the university, he focused on numerous mathematical topics,
such as the evaluation of limits for functions, integrals, and
derivatives; but did not focus on probability theory until his later
years.
Markov is known for his studies
and contributions to probability theory.
The most notable contribution is the development of a Markov
chain. According to Young
(1998), one can often predict
the
future status of some collection of random events if certain
information is available about the present status of the sequence of
events. As an example,
the movement of gas molecules in a container is random.
It can never be predicted exactly what any one molecule will do
at any given moment. But,
Markov showed, there are certain circumstances under which a later
state of molecules in the container can be predicted if certain
conditions exist among the molecules now.
The sequence of events under which this situation can occur is
called a Markov chain. (p. 335)
Although Markov found few
practical applications before he died in St. Petersburg in 1922, one can
find numerous applications of his work in biological sciences, physical
sciences, and technology.
Additional
Links:
MacTutor - Markov
The World Great
Mathematicians, Hong Kong Baptist University - Markov
Activity
#6:
Note:
As mentioned
in Markov’s biography above, a Markov Chain is a process in which an
event that happens is influenced by what happened immediately before
that event. Mathematically, a Markov Chain is "a sequence of
experiments, each if which results in one of a finite number of states
that we label 1, 2, . . . , m. If
pij is the probability of moving from state I to state j,
then the transition matrix P= [pij] of a Markov chain is the
m x n matrix
Notice
that the transition matrix P is a square matrix with entries that are
always between 0 and 1, inclusive, since they represent probabilities (Mizrahi
& Sullivan, 2000, p. 456).
Activity:
Voting Patterns (Mizrahi & Sullivan, 2000, p. 461):
The
voting pattern for a certain group of cities is such that 60% of the
Democratic (D) mayors were succeeded by Democrats and 40% by Republicans
(R). Also, 30% of the
Republican mayors were succeeded by Democrats and 70% by Republicans.
(a) Explain why the above is a Markov chain.
(b) Set up the 2 x 2 matrix P with columns and rows labeled D and R
to display these transitions.
(c) Compute P2 and P3.
Link
to solution.
Picture
reproduced from MacTutor History of Mathematics archive with
permission.
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