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Markov Chains

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Rebeca Padilla

on 6 June 2013

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Transcript of Markov Chains

Rebeca Padilla
22 May 2013 Markov Chains 1907 Andrei Andreyevian Markov began the study of an important new type of chance process.

In this process, the outcome of a given experiment can affect the outcome of the next experiment.

This work founded a completely new branch of probability theory. More Modern Probability Another Example: This new process is called a Markov Chain. Your turn... :) Born: June 14, 1856 in Ryazan, Russia
Died: July 20, 1922 in Petrograd (St. Petersburg), Russia Most of the time in probability we deal with independent trial processes, which are the basis of classical probability theory and much of statistics.

In these independent trial processes knowledge of the outcomes of the previous experiments does not influence our predictions for the outcomes of the next experiment. What is the probability of rolling a one? Six? Some things to remember... Probability Rocks! Thumbs up or thumbs down? Andrei Andreyevian Markov Markov graduated in 1878 from St. Petersburg University where he won the gold medal for submitting the best essay for the prize topic set by the faculty in that year - On the integration of differential equations by means of continued fractions.
He was awarded his Master's degree in 1880 for his thesis- On the binary quadratic forms with positive determinant.
After submitting his master's thesis, Markov began to teach at St. Petersburg University as a privatdozent while working for his doctorate .
He was awarded his doctorate in 1884 for his dissertation - On certain applications of continued fractions.
He published more than 120 scientific papers on number theory, continuous fraction theory, differential equations, probability theory, and statistics.
After 1900 Markov applied the method of counted fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory.
Remembered for Markov Chains. Auto Insurance Company Example: Weather Model Predicting the Weather Consider the following questions 1. If all of the auto insurance company's new policy holders represent the driving population in general, what will be the company's high/low risk distribution in one year from now? Two years from now?

2. What is the probability that a new customer will be in the high risk category two years from now? 5? 10? 20? 25?

3. If a customer is placed in the high risk category when they sign up, what is the probability that he/she will be in high or low risk 2 years from now? Hope so! It's GREEEAAT!!! New Perspective on Probability ? An auto insurance company places its policy holders into one of two categories; low risk&high risk. Policy holders can switch states when policy is renewed.

A person is high risk if they received a ticket within the past 12 months and low risk if they received zero tickets in the past 12 months.

Based on company data, a motorist that is currently at high risk has a 60% chance of being denoted high risk again when the policy renews and a 40% chance of being moved to low risk.

A motorist that is currently at low risk has a 15% chance of moving to the high risk category and a 35% chance of remaining in the low risk category.
Steady-State of the Weather Basketball Free Throws The coach of the middle school basketball team had the players practice their free throws for if and when they get fouled.

The coach analyzed the team data and learned that if the player makes the first shot, he/she is twice as likely to make the second shot as to miss.

If the player misses the first shot, he/she is three times as likely to miss the second shot.

Construct a transition matrix. Random Walk Other applications in the field of... physics
chemistry
testing
economics and finance
statistics
genetics Link to more applications
http://en.wikipedia.org/wiki/Markov_chain#Applications “The cases I indicated are NOT included in Bohlmann’s cases, but contain them as special cases. There is a huge difference. I am prepared to admit that Bohlmann gave an elegant special formula, but he did not point out even one new (after my article) case of the generalization of Bernoulli’s theorem”
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