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Stat 334

Lecture 11st May 2012

Dina DawoudM3 3126


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Course Outline Lectures: MWF 10:00-11:20 MC 4045 Tutorial: M 4:30-5:20 MC 4045

Notes: Stat 334 course notes available at Campus

Copy MC 2018

Supplementary lecture slides


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Final Grade: 20% Assignments + 30% Test +50% Final Exam

1 Test: 23 rd June 2012 4 Assignments: Dates can be found on course

outline on LEARN HELP:During tutorials and

Office Hours: M 3-4 and Tuesdays 12-1Held in M3 3126TA office Hours: TBA


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Motivation and Review

Multivariate distributions (discrete andcontinuous random variables, momentgenerating functions.)

Conditional Probability and Random Variables


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Markov Chain: Named after Andrey Markov, a Russian

Mathematician. It is a random process that possess what is

known as the memoryless property. Describes the evolution of a discrete process

over time, when time is discrete. The process randomly jumps from state to

state at each step. A step usually refers totime, but could also represent physicaldistance e.t.c


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As state changes (transitions) occur randomly,probabilities are associated with each possible

state change and these are called transitionprobabilities.

Example: Suppose that on class days you wake up and

only then decide whether to come to class. If you attended class the day before, you are

70% likely to go to class today, and if youskipped the class the day before, you are 80%likely to go today.


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Modeling this as a Markov Chain we can thenfind the probabilities that:

1. If you go to class on Monday, how likely areyou to go to class on Friday?

2. Assuming the class in infinitely long(Disaster!!!) approximately what portion of class will you attend?


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Poisson Process Named after the French Mathematician

Simeon-Denis Poisson.

His name can be found on the South-East sideof the Eiffel Tower in recognition of hiscontributions along with another 71 othernames!!


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A Poisson Process is a Stochastic processwhere events occur continuously and

independently (Continuous time, Discretestate space). Examples: Birth-death process: Is a special case of the

continuous Markov process. States: Currentpopulation size and the transitions are birth

and death. The number of goals in a soccer match. The arrival of customers in a queue.


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Brownian Motion The Wiener process is a continuous time

stochastic process named in honour of NorbertWiener.

It is often called standard Brownian motion, afterRobert Brown, who was a Botanist.

The model appeared after Robert Brownobserved the random movement of particlessuspended in a fluid (a liquid/ a gas).

The movement is described using a mathematicalmodel.

This model is used in several different fields and

one application is stock market fluctuations.


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It is prominent in Mathematical theory of Finance, in particular the Black-Scholes option

pricing formula. This formula was first introduced by Fischer

Black and Myron Scholes in 1973. However, Robert Merton was the 1 st to publish

the model. Robert Merton and Myron Scholes received

the 1997 Nobel prize in Economics. FischerBlack was mentioned for his contributions ashe had passed in 1995.


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Chapter 1: Motivation and Review Probability is a mathematical tool that is used to

quantify uncertainty and variability. (We canteliminate uncertainty)

We use mathematical models to try and predictfuture values and make decisions.

We cannot know if our model is correct.

Model is usually based on some prior knowledge.


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Example 1:You wish to purchase an insurance policy thatpays you $1 if a particular event A occurs beforethe years end.

What is a fair premium? Suppose the probability an event occurs is p

and X represents your return policy. X is called a random variable : A quantity that

takes on numerical values with specifiedprobabilities.


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In the example the Expected Value is the Fair

Premium .

We can also talk about Risk. To model risk, we

model the return on an uncertain investmentas a random variable, Y .


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Example 2: Bond Ratings: Bonds are rated every 3

months. Rating agencies rate bonds based on how

likely an issuer is to default. Suppose the possible ratings are:

(AAA, AA, A, BBB, BB, B, CCC, D)

Consider the following ratings migration model given inT. Schuermann et al., Credit Migration Matrices ,Encyclopaedia of Quantitative Risk Assessment, 2008, Wileyand Sons.


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We can use the rules of conditionalprobabilities to solve for questions such as:

What is the probability that the bond remainsrated at BBB after 6 months?


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