Chapter 10 - Chap 10 Random Walks and other applications...

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Unformatted text preview: Chap. 10 Random Walks and other applications 10.1 Random Walks L independent RV sequence n +1 n-1 p 1 q p = - Bernoulli trials 1 2 , , X X L L 1 2 n n = + + + S X X X L L = S h accumulated positive or negative excess at the n th trial L random walk model n takes a unit step or regular interval up or down n i X h h n S-u ” at the n th stepn 1 2 p q = = symmetricn unsymmetrical ¨ ” real life model Motion of gas molecules Thermal noise Stock value L model events Return to the origin The first return to the origin Waiting time for the first gain (first visit to 1 + ) Let n r = S represent the event “at stage n q the particle is at the point r ”n , n r P q 2G u { } , k n k n r n n P P r p q k- = = S @ @ k q successes n numbern n k- failures n number L ( ) 2 r k n k k n =-- =- 1 ( ) 2 k n r = + ( )/2 ( )/ 2 , ( ) / 2 n r n r n r n P p q n r +- = + (10-4) L binomial distributionn 0n 2 n r + integer between 0 and n q n q r even or odd Return to the origin n = S h i.e. r = in Eq.(10-4)n 2 n k = in Eq.(10-2) L n q even* : ” 2n th trial n { } 2 2 2 ( ) n n n n P pq u n = = S @ @ event n The Wiener Process n 1 2 ( ) n n nT = = + + + X S X X X L L random walk n take step size s q q { } i E = X h { } 2 2 i E s = X h { } ( ) E nT = X h { } 2 2 ( ) E nT ns = X ( Q independence of i X ) L 2 ( ) / 2 1 2 k n k k np npq n p q e k npq π--- 2245 for n q k in the npq vicinity of np q 1 2 p q = = 2 m k n =- { } 2 / 2 1 ( ) / 2 m n P nT ms e n π- = X ; 2 2 m n k = + 2 m k np- = m of the order of n q { } ( ) ( / ), i.e., Gaussian cdf with zero mean, P t ms G m n n σ = X ; L nT T t nT- < Wiener Process n n q q T q 2 s T α = define ( ) lim ( ) t t = W X h T q q t nT = Wiener Process n 2 / 2 1 ( , ) (0, ) 2 t f t e N t t ϖ α ϖ α πα- = = n 1st order statistics 10.2 Poisson Points and Shot noise L Poisson Points i t S : ” t : form n RV 1 t =- Z t h 1 t h t C : ” pointn Fig. 10-8 n n ( ) z f z e λ λ- = Z h ( ) 1 z F z e λ- = - Z h z L { } ( ) F z P z = Z Z { } ( , ) P t t z = + n (i.e., : ” ( ) t z + ) { } 1 ( , ) P t t z = - + = n ( ) 1 0! z e z λ λ- = - 1 z e λ- = - : ( ) z f z e λ λ- = Z L 1 t- =- w t h 1- t h t * pointn ( ) w f w e λ λ- = W h ( ) 1 w F w e λ- =- W h w L n n t =- X t h n t h t q n th pointn 1 ( ) ( 1)! n n x n f x x e n λ λ-- =- Gamma distribution Proofn { } { } ( ) 1 ( , ) n n F x P x P t t x n = = - + < X n 1 ( ) 1 ! k n x k x e k λ λ-- = = - : n q ( , ) t x C : ( ) n F x q x q 1 1 1 ( ) ( ) ( ) ( ) ! ! k k n n x n k k x k x f x e k k λ λ λ λ---- = = =- 1 2 ( ) ( ) ( ) ! ! k k n n x k k x x e k k λ λ λ λ--- = = =- 1 ( 1)! n n x x e n λ λ-- =- L 1 1 n n n n-- =- =- X X X t t (n points n )n ( ) x X f x e λ λ- = Poisson Points H ” modelsn Photon countn electron emissionn telephone callsn data communicationn visits to a doctorn arrivals at a park general form n Poisson Points n model n Given a sequence...
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Chapter 10 - Chap 10 Random Walks and other applications...

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