Optical Networks - _H_2 Random Processes_164

Optical Networks - _H_2 Random Processes_164 - 792 Random...

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Unformatted text preview: 792 Random Variables and Processes where r is a parameter associated with the distribution. It is easily verified that E [ X ] = r and σ 2 X = r . H.2 Random Processes Random processes are useful to model time-varying stochastic events. A random process X(t) is simply a sequence of random variables X(t 1 ), X(t 2 ), . . . , one for each instant of time. The first-order probability distribution function is given by F (x, t) = P { X(t) ≤ x } , and the first-order density function by f (x, t) = ∂F (x, t) ∂x . The second-order distribution function is the joint distribution function F (x 1 , x 2 , t 1 , t 2 ) = P { X(t 1 ) ≤ x 1 , X(t 2 ) ≤ x 2 } , and the corresponding second-order density function is defined as f (x 1 , x 2 , t 1 , t 2 ) = ∂ 2 F (x 1 , x 2 , t 1 , t 2 ) ∂x 1 ∂x 2 . The mean of the process is μ(t) = E [ X(t) ] = ∞ −∞ xf (x, t)dx. The autocorrelation of the process is R X (t 1 , t 2 ) = E [ X(t 1 )X(t 2 ) ] = ∞ −∞ ∞ −∞ x 1 x 2 f (x 1 , x...
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This note was uploaded on 01/15/2011 for the course ECE 6543 taught by Professor Boussert during the Spring '09 term at Georgia Tech.

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Optical Networks - _H_2 Random Processes_164 - 792 Random...

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