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Applied Probability III
APP MTH 3001/7056
Prof Nigel Bean
[email protected]
Room 6.20
Ingkarni Wardli
Course outline
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Section 00:
Probability Review
(Assumed Knowledge)
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Section 01:
Introduction
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Section 02:
Probability and measure
◆
Sample space
◆
Algebras and
σ
-algebras of events
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Probability measure
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Section 03:
Discrete Time Markov Chains
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Basic definitions
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Hitting probabilities and hitting times
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Classification of states
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Recurrence and transience
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Limiting behaviour
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Section 04:
Martingales
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Definition of a Martingale
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Stopping times and the Optional stopping theorem
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Section 05:
Brownian motion

3 / 326Example 0.1:Consider an experiment which consists of tossing a fair coin twice.Then the sample space isΩ ={ω1, ω2, ω3, ω4}={HH, HT, TH, TT}.Events4 / 326Here are some examples of events■the event that a “head” comes up when a coin is tossed,■the event that two “heads” come up when a coin is tossed twice,■the event that a particular candidate wins an election,■the event that the sun rises tomorrow.In general, ifAandBare events, they can be thought of as subsets of the samplespaceΩ.That is, we haveA, B⊆Ω.The event that at least one ofAorBoccurs is called theunionofAandB, andis writtenA∪B.The event that both occur is called theintersectionofAandB, and is writtenA∩B.This notation extends to all countable collections of eventsAi, i= 1,2, . . ., forwhich we writeA1∪A2∪ · · ·=∪∞i=1Ai,A1∩A2∩ · · ·=∩∞i=1Ai.