Stochastic Processes

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1 Stochastic Processes Review o Elemetar Probabilit Lecture I Hamid R. Rabiee Fall 20 Ali Jalali

2 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

3 Histor & Philosoph Started b gamblers dispute Probabilit as a game aalzer! Formulated b B. Pascal ad P. Fermet First Problem (654) : Double Si durig 24 throws First Book (657) : Christia Huges De Ratiociiis i Ludo Aleae I Germa 657.

4 Histor & Philosoph (Cot d) Rapid developmet durig 8 th Cetur Major Cotributios: J. Beroulli ( ) A. De Moivre ( )

5 Histor & Philosoph (Cot d) A reaissace: Geeralizig the cocepts rom mathematical aalsis o games to aalzig scietiic ad practical problems: P. Laplace ( ) New approach irst book: P. Laplace Théorie Aaltique des Probabilités I Frace 82.

6 Histor & Philosoph (Cot d) 9 th cetur s developmets: Theor o errors Actuarial mathematics Statistical mechaics Other giats i the ield: Chebshev Markov ad Kolmogorov

7 Histor & Philosoph (Cot d) Moder theor o probabilit (20 th ) : A. Kolmogorov : Aiomatic approach First moder book: A. Kolmogorov Foudatios o Probabilit Theor Chelsea New ork 950 Nowadas Probabilit theor as a part o a theor called Measure theor!

8 Histor & Philosoph (Cot d) Two major philosophies: Frequetist Philosoph Observatio is eough Baesia Philosoph Observatio is NOT eough Prior kowledge is essetial Both are useul

9 Histor & Philosoph (Cot d) Frequetist philosoph There eist ied parameters like mea. There is a uderlig distributio rom which samples are draw Likelihood uctios(l()) maimize parameter/data For Gaussia distributio the L() or the mea happes to be /N i i or the average. Baesia philosoph Parameters are variable Variatio o the parameter deied b the prior probabilit This is combied with sample data p(/) to update the posterior distributio p(/). Mea o the posterior p(/)ca be cosidered a poit estimate o.

10 Histor & Philosoph (Cot d) A Eample: A coi is tossed 000 times ieldig 800 heads ad 200 tails. Let p = P(heads) be the bias o the coi. What is p? Baesia Aalsis Our prior kowledge (believe) : p (Uiorm(0)) Our posterior kowledge : pobservatio p 800 p Frequetist Aalsis Aswer is a estimator pˆ such that 8 Mea : E pˆ 0. Coidece Iterval : P pˆ

11 Histor & Philosoph (Cot d) Further readig: hist/stathist.htm stat/histor/idehistor.shtml istprob.pd

12 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

13 Radom Variables F P F F P P

14 Radom Variables (Cot d) Radom variable is a uctio ( mappig ) rom a set o possible outcomes o the eperimet to a iterval o real (comple) umbers. I other words : Outcomes F I R : : F r I Real Lie

15 Radom Variables (Cot d) Eample I : Mappig aces o a dice to the irst si atural umbers. Eample II : Mappig height o a ma to the real iterval (03] (meter or somethig else). Eample III : Mappig success i a eam to the discrete iterval [020] b quatum 0..

16 Radom Variables (Cot d) Radom Variables Discrete Dice Coi Grade o a course etc. Cotiuous Temperature Humidit Legth etc. Radom Variables Real Comple

17 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

18 Desit/Distributio Fuctios Probabilit Mass Fuctio (PMF) Discrete radom variables Summatio o impulses The magitude o each impulse represets the probabilit o occurrece o the outcome P Eample I: Rollig a air dice PMF 6 6 i i

19 Desit/Distributio Fuctios (Cot d) Eample II: Summatio o two air dices P Note : Summatio o all probabilities should be equal to ONE. (Wh?)

20 Desit/Distributio Fuctios (Cot d) Probabilit Desit Fuctio (PDF) Cotiuous radom variables The probabilit o occurrece o will be P d. 0 d 2 d 2 P P

21 Desit/Distributio Fuctios (Cot d) Some amous masses ad desities Uiorm Desit P a. U ed Ubegi a Gaussia (Normal) Desit. 2 P a. e 2 2. N 2 2

22 Desit/Distributio Fuctios (Cot d) Biomial Desit Poisso Desit! : Note e p N p N.. 0 N.p N Importat Fact:!... :. p N e p p N llarge N For Suiciet p N N

23 Desit/Distributio Fuctios (Cot d) Cauch Desit P 2 2 Weibull Desit k k e k

24 Desit/Distributio Fuctios (Cot d) Epoetial Desit Raleigh Desit e U e e

25 Desit/Distributio Fuctios (Cot d) Epected Value The most likelihood value Liear Operator Fuctio o a radom variable Epectatio E E E. d a. b a. E b g g. d

26 Desit/Distributio Fuctios (Cot d) PDF o a uctio o radom variables Assume RV such that g The iverse equatio g ma have more tha oe solutio called... 2 PDF o ca be obtaied rom PDF o as ollows i d d g i i

27 Desit/Distributio Fuctios (Cot d) Cumulative Distributio Fuctio (CDF) Both Cotiuous ad Discrete Could be deied as the itegratio o PDF PDF CDF F F P. d CDF()

28 Desit/Distributio Fuctios (Cot d) Some CDF properties No-decreasig Right Cotiuous F(-iiit) = 0 F(iiit) =

29 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

30 Joit/Coditioal Distributios Joit Probabilit Fuctios Desit Distributio Eample I I a rollig air dice eperimet represet the outcome as a 3-bit digital umber z... 0 ; 6 0 ; 3 0; 3 0 0; 6 O W z dd ad P F

31 Joit/Coditioal Distributios (Cot d) Eample II Two ormal radom variables What is r? Idepedet Evets (Strog Aiom) r r e r

32 Joit/Coditioal Distributios (Cot d) Obtaiig oe variable desit uctios Distributio uctios ca be obtaied just rom the desit uctios. (How?) d d

33 Joit/Coditioal Distributios (Cot d) Coditioal Desit Fuctio Probabilit o occurrece o a evet i aother evet is observed (we kow what is). Baes Rule d..

34 Joit/Coditioal Distributios (Cot d) Eample I Rollig a air dice : the outcome is a eve umber : the outcome is a prime umber Eample II Joit ormal (Gaussia) radom variables P P P r r e r

35 Joit/Coditioal Distributios (Cot d) Coditioal Distributio Fuctio Note that is a costat durig the itegratio. dt t dt t d while P F

36 Joit/Coditioal Distributios (Cot d) Idepedet Radom Variables Remember! Idepedec is NOT heuristic..

37 Joit/Coditioal Distributios (Cot d) PDF o a uctios o joit radom variables Assume that The iverse equatio set has a set o solutios Deie Jacobea matri as ollows The joit PDF will be g V U ) ( U V g ) ( V U V U J i i i U V i i J determiat absolute v u.

38 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

39 Correlatio Kowig about a radom variable how much iormatio will we gai about the other radom variable? Shows liear similarit More ormal: E Crr. Covariace is ormalized correlatio. E.. Cov( ) E

40 Correlatio (cot d) Variace Covariace o a radom variable with itsel Var 2 E Relatio betwee correlatio ad covariace Stadard Deviatio Square root o variace E

41 Correlatio (cot d) Momets th order momet o a radom variable is the epected value o M E Normalized orm M E Mea is irst momet Variace is secod momet added b the square o the mea

42 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems

43 Importat Theorems Cetral limit theorem Suppose i.i.d. (Idepedet Ideticall Distributed) RVs k with iite variaces Let S i a. PDF o S coverges to a ormal distributio as icreases regardless to the desit o RVs. Eceptio : Cauch Distributio (Wh?)

44 Importat Theorems (cot d) Law o Large Numbers (Weak) For i.i.d. RVs k 0 Pr lim 0 i i

45 Importat Theorems (cot d) Law o Large Numbers (Strog) For i.i.d. RVs k Wh this deiitio is stroger tha beore? lim Pr i i

46 Importat Theorems (cot d) Chebshev s Iequalit Let be a oegative RV Let c be a positive umber Aother orm: Pr Pr c c E 2 2 It could be rewritte or egative RVs. (How?)

47 Importat Theorems (cot d) Schwarz Iequalit For two RVs ad with iite secod momets E E.E Equalit holds i case o liear depedec.

48 Net Lecture Elemets o Stochastic Processes

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