Notes - Applied Probability III APP MTH 3001\/7056 Prof Nigel Bean [email protected] Room 6.20 Ingkarni Wardli 1 326 Course outline Section 00

# Notes - Applied Probability III APP MTH 3001/7056 Prof...

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1 / 326 Applied Probability III APP MTH 3001/7056 Prof Nigel Bean [email protected] Room 6.20 Ingkarni Wardli Course outline 2 / 326 Section 00: Probability Review (Assumed Knowledge) Section 01: Introduction Section 02: Probability and measure Sample space Algebras and σ -algebras of events Probability measure Section 03: Discrete Time Markov Chains Basic definitions Hitting probabilities and hitting times Classification of states Recurrence and transience Limiting behaviour Section 04: Martingales Definition of a Martingale Stopping times and the Optional stopping theorem 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 eventsthe 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, BΩ.The event that at least one ofAorBoccurs is called theunionofAandB, andis writtenAB.The event that both occur is called theintersectionofAandB, and is writtenAB.This notation extends to all countable collections of eventsAi, i= 1,2, . . ., forwhich we writeA1A2∪ · · ·=i=1Ai,A1A2∩ · · ·=i=1Ai.
Events and probability 5 / 326 The probability of an event A is written P ( A ) . The certain event, denoted by Ω , always occurs, so P (Ω) = 1 The impossible event, denoted by , never occurs, so P ( ) = 0 . It is always the case that 0 P ( A ) 1 for any event A , so that if you ever calculate a probability and it is negative or greater than one, then you know you have made a mistake ! Definition 0.1: Disjoint events Events A and B are said to be disjoint if A B = . That is, if A and B cannot both occur. For disjoint events A and B , we have P ( A B ) = P ( A ) + P ( B ) . A more general statement of the addition law is as follows. For events A i , i = 1 , 2 , . . . , with A j and A k disjoint for all j = k , we have P ( i =1 A i ) = i =1 P ( A i ) . (0.1) Conditional probability 6 / 326 Many of the probabilities we will encounter in this course, and indeed many statements about chance in general, are conditional probabilities, taking the form The probability of A , given that B occurs ”, where A and B are events. Definition 0.2: Conditional probability If P ( B ) > 0 then the conditional probability that A occurs given that B occurs is defined to be P A B = P ( A B ) P ( B ) . (0.2) Note: For event