STA 348 Introduction to Stochastic Processes. Lecture 1

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1 STA 348 Itroductio to Stochastic Processes Lecture 1 1

2 Admiis-trivia Istructor: Sotirios Damouras Proouced Sho-tee-ree-os or Sam Cotact Ifo: Office hours: SE/DV 4062, every Mo 2-4pm ad Tues 4-5pm, or by appoitmet ( ) Course web page: (UofT Portal) All course material (outlie, lecture slides, assigmets & solutios) posted o portal

3 Outlie Textbook: Itroductio to probability models 10 th Ed, by Sheldo M. Ross (i bookstore) Cover (parts of) 1-8, & extra topics if time permits Evaluatio: 9 Weekly Assigmets, best 8/9 worth 20% start of tutorial, NO late submissios 2 Term Tests, worth 20% each NO make-up tests. Weight shifted to Fial exam with UofT Medical ote AND absece declaratio o ROSI Fial Exam, worth 40%

4 Importat Dates LEC (TUE IB 220) LEC (THU IB 200) TUT (FRI IB 200) Sept o tutorial assig 1 due assig 2 due assig 3 due Oct assig 4 due 11 Midterm o tutorial assig 5 due assig 6 due Nov assig 7 due 8 Midterm o tutorial assig 8 due assig 9 due 4

5 What Is This Course About? Modelig & aalyzig behavior of a collectio of depedet radom variables (RV s) (X 1,X 2, ) = {X t } t=1,2, is a Stochastic Process How is this differet from Statistics? Statistics: X 1, X 2, is idepedet radom sample from some distributio/populatio Stochastic Processes: { X t } t=1,2, is collectio of depedet RV s, describig a radom process at differet poits t=1, 2, i time or space E.g. Coutry s populatio at year t = 1, 2, 5

6 Example 1 Gamble $10 i Roulette (bettig o red / black) till you double or loose it If you wi bet o red/black, you double bet amout P(wiig idividual bet) = 18/(36+2) =.4737 Which is the best strategy for maximizig the chace of doublig moey (reachig $20): A. Bet $10 all at oce B. Bet $1 at a time C. It does t matter 6

7 Example 2 You are tossig a fair coi, i.e. P(Heads) = P(Tails) = ½, ad coutig the # of tosses till oe of two patters occurs Patter 1 = (H,H) & Patter 2 = (H,T) Which patter appears first o average? A. Patter 1 B. Patter 2 C. Both are equally likely 7

8 Example 3 A type of bacterium reproduces i the followig way: With prob. ½ it splits ito 2 idetical copies With prob. ½ it dies before dividig If you place 10 such bacteria o a Petri dish, what happes to their (log-ru) populatio A. It will certaily survive idefiitely B. It will certaily die out evetually C. It will ca either survive or die (w/ some prob s) 8

If rate λ doubles, how should rate μ chage so that the mea time a perso stays i

9 Example 4 Cosider service queue (e.g. airport security) People arrive at rate λ, ad People get served at rate μ (λ < μ) If rate λ doubles, how should rate μ chage so that the mea time a perso stays i the system (wait + service time) stays the same? A. μ should double B. μ should less tha double C. μ should more tha double 9

10 Stochastic Processes How to aalyze collectio of RV s? E.g. {X t } t=1,2, with joit pdf If RV s are idepedet work with margials f x, x,... f x f x... 1,2, If RV s are depedet, work with coditioals 1,2, , , Stochastic Processes mostly deal with various types of coditioal depedece But first, eed to brush up our Probability Theory f x, x,... 1,2, (as i Stats) f x, x,... f x x,... f x x,

11 Experimets & Evets Experimet: process with radom result Outcome: elemetary result of experimet Sample Space (S): Set of all outcomes Evet: A arbitrary collectio of outcomes Evets are subsets of S, deoted by capital letters E.g. Rollig a 6-sided die, E = {eve roll} = {2,4,6} Ve Diagram: outcomes S evet E 11

12 Combiig Evets Uio: AB A B { or } A A U B B A A A A i1 i 1 2 Itersectio: A B A B { A ad B} A A B B A A A A i1 i 1 2 Complemet: c A { ot A} A A C 12

13 De Morga s Laws c c c A B A B A B C A B c c c A B A B A B C A B More geerally: c c c c i1 Ai A1 A2 A c c c c i1 Ai A1 A2 A 13

14 Probabilities Cosider a experimet with sample space S. A probability (measure) is a fuctio P( ) that assigs umbers P(A) to evets A S, so that: P A P S If A, A, A, are mutually exclusive evets, the P i Ai P Ai i1 Evets A, A, are mutually exclusive if A A for all i j i j 14

15 Coditioal Probability & Idepedece Coditioal Probability: P(A B) is probability of evet A give that evet B has occurred P A B P A B, for PB 0 P B Idepedece: Evets A, B are idepedet if P A B P( A) P A B P A PB P B A P( B) 15

16 Mutual Idepedece Geeralizatio to 2 evets: A fiite collectio of evets 1 2 A, A,, A : Pairwise idep. does ot imply mutual idep. 1 2 is called ( mutually ) idepedet if for ay sub-collectio k k k m P A A P A P A, i j i j i j 1 2 A, A,, A A, A,, A are mutually idepedet m m i k i1 k P A P A 1 i i 16

17 Example S = (H,H) (H,T) (T,H) (T,T) Cosider flippig two fair cois & defie evets A={(H,H),(H,T)}, B={(H,H),(T,H)}, C={(H,H),(T,T)} Are A, B, ad C pairwise idepedet? Are A, B, ad C mutually idepedet? 17

18 Rules of Probability Complemet Rule: Additio Rule: If A, B mutually exclusive, the P( A B) P( A) P( B) Multiplicatio Rule: c P( A ) 1 P( A) A B P( A B) P( A) P( B) P( A B) P( A B) P( A B) P( B) P( B A) P( A) A B If A, B idepedet, the P( AB) P( A) P( B) 18

19 Rules of Probability Geeralizatios for 2 evets: For ay fiite collectio of evets {A 1,A 2,...,A } Additio Rule: i1 i i1 i ij i j P A P A P A A Multiplicatio Rule: 1 1 i j k 1 i P A A A P A ijk i i1 i P A P A P A A P A A A P A A A A

20 Law of Total Probability Partitio of S is fiite set of evets {B 1,B 2,...,B }, such that: B B, i j & B S i j i1 i For ay evet A ad partitio {B 1,B 2,...,B }, S B 2 B 1 A B 3 P A P A B P B P A B i1 i i1 i i From additio rule, sice: A A B1 A B2 A B ad A B i A Bj, i j 20

21 Bayes Formula Let {B 1,B 2,...,B } be a partitio of S such that P(B i )>0, for i=1,2,...,. The, for ay evet A P B j A P A B P B i 1 j P A B P B i j i For =2: P B A P( B) P( A B) c c P( B) P( A B) P( B ) P( A B ) Method for revisig evet B j s probability, give iformatio o occurrece of aother evet A Kow: P(B j ) prior probability, P(A B i ), i=1,, Wat: P(B j A) posterior probability 21

22 Coutig Rules Permutatio Rule: Number of permutatios of r objects, selected w/o repeats from objects:! Pr ( 1) ( r), where 0 r r! Combiatio Rule: Number of combiatios of r objects, selected w/o repeats from objects: Pr! Cr, where 0 r r r! r! r! Biomial Theorem: i0 x y x y i i i 22

23 Example (Matchig Problem: 1-Q32) At a party, # people get druk & o their way out they grab a coat at radom. What is the probability that obody got their ow coat? 23

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25 Example # poits are radomly draw o a circle. What is the probability that all poits lie i a semi-circle? 25

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What is Probability?

Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability