Infinitesimal Calculus of Random Walk and Poisson Processes

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Gauge Institute Journal, H. Vic Dannon Var[ Bi] = E[ Bi ] − ( E[ B ]) ( ) i = dx 2 ( dx ) 2 2 0 2 . 10.2 The Binomial Distribution of the Walk B( ζ , t) = B + B + ... + BN 1 2 is a Random Process with EB [ ( ζ , t)] = 0, distributed Binomially Var[ B( ζ , t)] = N( dx) , ⎛ Pr ( , ) N ⎞ − ≤ ≤ + = ⎜ ⎟ N ( x 1dx B t x 1dx) ζ 1 2 2 ⎜ M+ N ⎟ 2 ⎜⎝ 2 ⎠ 2 Proof: Since the B i are independent, EB [ ( ζ , t)] = EB [ ] + ... + EB [ N ] = 0 1 0 0 Var[ B( ζ , t)] = Var[ B ] + ... + Var[ BN ] = N( dx) 1 ( dx ) ( dx ) 2 2 B(,) ζ t has a Binomial distribution, ⎛N ⎞ 1 1 Pr ( x − dx ≤ X( ζ, t) ≤ x + dx) = p q 2 2 ⎜K ⎜⎝ ⎠⎟ 2 K N−K , 38
Gauge Institute Journal, H. Vic Dannon From N = K +L , ⎛N ⎞ = ⎜K ⎜⎝ ⎠⎟ ⎛N ⎞ = ⎜K ⎜⎝ ⎠⎟ K 1 1 ( ) ( ) 1 2 N 2 2 . N−K , we have M = K − L, K N+ M 2 = , L = . N−M 2 Thus, ⎛ Pr ( , ) N ⎞ − ≤ ≤ + = ⎜ ⎟ 1 . N 1 1 ( x dx B t x dx) ζ 2 2 ⎜ M+ N 2 ⎜⎝ ⎟ 2 ⎠ 10.3 The Gaussian Distribution of the Walk If 2 ( dx) = 2 D( dt) , where the Drift Coefficient D is a constant Then, the Binomial distribution of B(,) ζ t is infinitesimally close to a Gaussian distribution of a Random Signal with μ = 0, σ = t2D = Ndx. fxt (,) ≈ 1 e 2π t2D 2 1 x − 2t2D 39
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