Stochastic Processes. Review of Elementary Probability Lecture I. Hamid R. Rabiee Ali Jalali

Transcription

1 Stochastic Processes Review o Elementary Probability bili Lecture I Hamid R. Rabiee Ali Jalali

2 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

3 History & Philosophy Started by gamblers dispute Probability as a game analyzer! Formulated by B. Pascal and P. Fermet First Problem (654) : Double Si during 4 throws First Book (657) : Christian Huygens, De Ratiociniis in Ludo Aleae, In German, 657.

4 History & Philosophy (Cont d) Rapid development during 8 th Century Major Contributions: J. Bernoulli ( ) A. De Moivre ( )

5 History & Philosophy (Cont d) A renaissance: Generalizing the concepts rom mathematical analysis o games to analyzing scientiic and practical problems: P. Laplace (749-87) 87) New approach irst book: P. Laplace, Théorie Analytique des Probabilités, In France, 8.

6 History & Philosophy (Cont d) 9 th century s developments: Theory o errors Actuarial mathematics Statisticalti ti mechanics Other giants in the ield: Chebyshev, Markov and Kolmogorov

7 History & Philosophy (Cont d) Modern theory o probability (0 th ) : A. Kolmogorov : Aiomatic approach First modern book: A. Kolmogorov, Foundations o Probability Theory, Chelsea, New York, 950 Nowadays, Probability theory as a part o a theory called Measure theory!

8 History & Philosophy (Cont d) Two major philosophies: Frequentist Philosophy Observation is enough Bayesian Philosophy Observation is NOT enough Prior knowledge is essential Both are useul

9 History & Philosophy (Cont d) Frequentist philosophy There eist ied parameters like mean,. There is an underlying distribution rom which samples are drawn Likelihood unctions(l()) maimize parameter/data For Gaussian distribution the L() or the mean happens to be /N i i or the average. Bayesian philosophy Parameters are variable Variation o the parameter deined by the prior probability This is combined with sample data p(/) to update the posterior distribution p(/). Mean o the posterior, p(/),can be considered a point estimate o.

10 History & Philosophy (Cont d) An Eample: A coin is tossed 000 times, yielding 800 heads and 00 tails. Let p = P(heads) be the bias o the coin. What is p? Bayesian Analysis Our prior knowledge (believe) : p = (Uniorm(0,)) Our posterior knowledge : p Observation = p 800 p Frequentist Analysis Answer is an estimator pˆ such that [ ˆ ] = 0. 8 Mean : E p Conidence Interval : P ( ) ( ) ( ) 00 ( pˆ 0.86) 0. 95

11 History & Philosophy (Cont d) Further reading: hist/stathist.htm t thi t ht stat/history/indehistory.shtml istprob.pd

12 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

13 Random Variables Probability Space A triple o (, F, P) represents a nonempty set, whose elements are sometimes known as outcomes or states o nature F represents a set, whose elements are called events. The events are subsets o. F should be a Borel Field. P represents the probability measure. Fact: P( ) =

14 Random Variables (Cont d) Random variable is a unction ( mapping ) rom a set o possible outcomes o the eperiment to an interval o real (comple) numbers. In other words : F : F I : I R ( ) = r

15 Random Variables (Cont d) Eample I : Mapping aces o a dice to the irst si natural numbers. Eample II : Mapping height o a man to the real interval (0,3] (meter or something else). Eample III : Mapping success in an eam to the discrete interval [0,0] by quantum 0..

16 Random Variables (Cont d) Random Variables Discrete Dice, Coin, Grade o a course, etc. Continuous Temperature, Humidity, Length, etc. Random Variables Real Comple

17 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

18 Density/Distribution Functions Probability Mass Function (PMF) Discrete random variables Summation o impulses The magnitude o each impulse represents the probability o occurrence o the outcome P( ) Eample I: Rolling a air dice 6 PMF = 6 6 ( i ) i =

19 Density/Distribution Functions (Cont d) Eample II: 6 Summation o two air dices P( ) Note : Summation o all probabilities should be equal to ONE. (Why?)

20 Density/Distribution Functions (Cont d) Probability Density Function (PDF) Continuous random variables The probability o occurrence o will be P ( ). d 0 d, + d P( ) P( )

21 Density/Distribution Functions (Cont d) Some amous masses and densities Uniorm Density P( ) a ( ) =.( U ( end ) U ( begin) ) a Gaussian (Normal) Density P( ) a. ( ) =. e ( µ ). = N ( µ, ) µ

22 Density/Distribution Functions (Cont d) Binomial Density ( n) N n ( n) ( p) n p N =.. n Poisson Density ( ) 0 N.p N n ( ) = e!! Note : ( + ) ( + ) =! Important Fact: For Suicient ly large N :. ( p)! N n N n. p n " e N. p ( N. p) n! n

23 Density/Distribution Functions (Cont d) Cauchy Density ( ) ( ) # µ # + = ( ) P Weibull Density ( ) # µ ( ) k k e k =!!!

24 Density/Distribution Functions (Cont d) Eponential Density ( ) ( ) < = = e U e!!!! Rayleigh Density < 0 0 ( ). e =

25 Density/Distribution Functions (Cont d) Epected Value The most likelihood value E Linear Operator ' & ' [ ]. ( ) = d Function o a random variable Epectation [ a. + b] = a. E[ ] b E + E ' & ' [ g( )] g( ). ( ) = d

26 Density/Distribution Functions (Cont d) PDF o a unction o random variables Assume RV Y such that Y = g( ) The inverse equation = g ( Y ) may have more than one solution called,,..., PDF o Y can be obtained rom PDF o as ollows Y ( y) = n i= d d g ( ) i ( ) n = i

27 Density/Distribution Functions (Cont d) Cumulative Distribution Function (CDF) Both Continuous and Discrete Could be deined as the integration o PDF PDF ( ) CDF ( ) = F ( ) = P( ) & ' ( ) = ( ) F. d

28 Density/Distribution Functions (Cont d) Some CDF properties Non-decreasing Right Continuous F(-ininity) it = 0 F(ininity) =

29 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

30 Joint/Conditional Distributions Joint Probability Functions Density Distribution Eample I F, Y (, y) = P( and Y y) = y & &, Y (, y) dyd In a rolling air dice eperiment represent the outcome as a 3-bit digital number yz., Y (, y) ( 6 ( ( 3 = 3 ( ( 6 ( ( 0 = 0; y = 0 = 0; y = = ; y = 0 = ; y = O. W. ' ' yz

31 Joint/Conditional Distributions (Cont d) Eample II Two normal random variables ( ) ( ) ( ) ( ) ( )( ) + = y y y y y r y r Y e y µ µ µ µ., What is r? Independent Events (Strong Aiom) ( ) y Y r y,... ( ) ( ) ( ) y y Y Y.,, =

32 Joint/Conditional Distributions (Cont d) Obtaining one variable density unctions ' ( ) = &, Y ( y), ' ' ( y ) = &, Y ( y ) dy = d Y, ' Distribution unctions can be obtained just rom the density unctions. (How?)

33 Joint/Conditional Distributions (Cont d) Conditional Density Function Probability o occurrence o an event i another event is observed (we know what Y is). Y ( y) = (, y ) ( y), Y, Y Bayes Rule Y ( y) = ' & ' Y Y ( y ). ( ) ( y ). ( ) d

34 Joint/Conditional Distributions (Cont d) Eample I Rolling a air dice : the outcome is an even number Y : the outcome is a prime number Eample II Joint normal (Gaussian) random variables ( ) ( ) ( ) 3 6, = = = P Y Y P Y P ( ) ( ) =.. y y y r r Y e r y µ µ

35 Joint/Conditional Distributions (Cont d) Conditional Distribution Function F Y ( y) = P( while Y = y) = = & ' & ' ' & ' Y, Y, Y ( y) ( t, y) ( t, y) d dt dt Note that y is a constant during the integration.

36 Joint/Conditional Distributions (Cont d) Independent Random Variables ( y) Y = = =, Y (, y) Y ( y) ( ). Y ( y) Y ( y) ( ) Remember! Independency is NOT heuristic.

37 Joint/Conditional Distributions (Cont d) PDF o a unctions o joint random variables Assume that ( U, V ) = g(, Y ) The inverse equation set (, Y ) = g ( U, V ) has a set o solutions (, Y ), (, Y ),..., ( n, Y n ) Deine Jacobean matri as ollows The joint PDF will be U, V ( u, v) = n absolute, Y (, y ) determinant J, - U - V - - = / - U - ( ) J (, y) ( y ). i= =, i i i i - V -Y * ) ) ) ).

38 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

39 Correlation Knowing about a random variable, how much inormation will we gain about the other random variable Y? Shows linear similarity More ormal: (, Y ) = E[ Y ] Crr. Covariance is normalized correlation [( µ )(. Y µ Y )] = E[. Y ] µ. Y Cov (, Y ) = E µ

40 Correlation (cont d) Variance Covariance o a random variable with itsel [ ] ( ) = = E ( ) Var µ Relation between correlation and covariance [ ] = + E µ Standard Deviation Square root o variance

41 Correlation (cont d) Moments n th order moment o a random variable is the epected value o n M = E n ( n ) Normalized orm n ( ) n ) M = E µ Mean is irst moment Variance is second moment added by the square o the mean

42 Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions Correlation Important Theorems

43 Important Theorems Central limit theorem Suppose i.i.d. (Independent Identically Distributed) RVs k with inite variances Let n = n S a. i= n n PDF o S n converges to a normal distribution as n increases, regardless to the density o RVs. Eception : Cauchy Distribution (Why?)

44 Important Theorems (cont d) Law o Large Numbers (Weak) For i.i.d. RVs k 4 n ( i i= lim n' Pr µ > ( n ( 5 > 0 5 = 0 ( ( ( 0

45 Important Theorems (cont d) Law o Large Numbers (Strong) For i.i.d. RVs k 0 n ( i ( i= Pr limn' = µ = ( n ( ( ( Why this deinition is stronger than beore?

46 Important Theorems (cont d) Chebyshev s Inequality Let be a nonnegative RV Let c be a positive number Another orm: Pr Pr c { > c } E [ ] { µ > 5} 5 It could be rewritten or negative RVs. (How?)

47 Important Theorems (cont d) Schwarz Inequality For two RVs and Y with inite second moments E [ ] [ ] [ ]. Y E.E Y Equality holds in case o linear dependency.

48 Net Lecture Elements o Stochastic Processes