Chapter 10 - 1 Chapter 10 Random Walks and Other...

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Unformatted text preview: 1 Chapter 10 Random Walks and Other Applications Teacher: Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 10-1 Random Walks h& independent RV sequence 1 +1 1 -1 1 p 1 1 q p = - Bernoulli trials 1 2 , , X X 1 2 n n = + + + S X X X = S S S accumulated positive or negative excess at the n th trial h random walk model 1 takes a unit step or regular interval up or down 1 i X S S n S S S g / at the n th step1 1 2 p q = = symmetric1 unsymmetrical 3 T b& real lifeb model Motion of gas molecules Thermal noise Stock value h model b events Return to the origin The first return to the origin Waiting time for the first gain (first visit to 1 + ) 4 Let n r = S represent the event at stage nq the particle is at the point r 1 , n r P { } , k n k n r n n P P r p q k- = = S S 1 k q successes 1 number1 n k- failures 1 number h ( ) 2 r k n k k n =-- =- 1 ( ) 2 k n r = + 1 ( )/ 2 ( )/ 2 , ( )/ 2 n r n r n r n P p q n r +- = + (10-4) h binomial distribution1 01 2 n r + integer between and n q n q r even or odd 5 Return to the origin n = S S i.e. r = in Eq.(10-4)1 2 n k = in Eq.(10-2) h n q evenj 2n th trial 1 { } 2 2 2 ( ) n n n n P pq u n = = S S S event 1 1 6 The Wiener Process 1 1 2 ( ) n n nT = = + + + X S X X X random walk 1 take step size s { } i E = X S { } 2 2 i E s = X { } ( ) E nT = X S { } 2 2 ( ) E nT ns = X ( 1 independence of i X ) 7 h& 2 ( ) /2 1 2 k n k k np npq n p q e k npq --- 2245 for n q k in the npq vicinity of npq 1 2 p q = = 2 m k n =- (10-2)1 { } 2 / 2 1 ( ) / 2 m n P nT ms e n - = X S 2 2 m n k = + 2 m k np- = m of the order of n { } ( ) ( / ) P t ms G m n X S for nT T t nT- < i.e., Gaussian cdf with zero mean, n = 8 Wiener Process 1 n T 2 s T = define ( ) lim ( ) t t = W X S T t nT = Wiener Process 1 2 /2 1 ( , ) (0, ) 2 t f t e N t t - = = 1 1st order statistics 9 10.2 Poisson Points and Shot noise h Poisson Points i t t form 1 RV 1 t =- Z t S 1 t S t q point1 Fig. 10-8 1 1 ( ) z f z e - = Z S ( ) 1 z F z e - =- Z S z 10 h& & { } ( ) 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 - = - 1 z e - = - ( ) z f z e - = Z 11 h& & 1 t- =- w t S 1- t S t point1 ( ) w f w e - = W S ( ) 1 w F w e - =- W S w h n n t =- X t S n t S t q n th point1 1 ( ) ( 1)! n n x n f x x e n -- =- Gamma distribution 12 Proof1 { } { } ( ) 1 ( , ) n n F x P x P t t x n = = - + < X n 1 ( ) 1 !...
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Chapter 10 - 1 Chapter 10 Random Walks and Other...

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