07Stochastic

07Stochastic - MAE 591 RANDOM DATA Stochastic Processes A...

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MAE 591 RANDOM DATA Stochastic Processes A stochastic process x is statistically determined if the N -th order probability distribution function t i () ( ) Px t x t x t xt N N N N N (() ,() , , () )P r { 1 1 2 21 1 12 2 2 ⋅⋅⋅⋅ } = ∩⋅⋅⋅⋅⋅ ∩≤ or N -th order probability density function px t x t N N , () 1 1 2 2 ) ⋅⋅⋅⋅ is known for any tt t N ,, , ⋅⋅⋅⋅⋅ . Now let us consider the expected value ( Ensemble average ) of a real single-valued continuous function of the random variables , gx x (, ) ( ) and ( ) Egxx gxxpxxd xd x gxx [ (,) ] == −∞ −∞ 1. Mean : g xx = µ x t Ext xt pxt dx [ () ] () ( () ) −∞ 2. Mean Square = ( : gx ) rms 2 x = 2 Ψ x tE x t x t p x t d 222 [ ] ) x −∞ 3. Variance = σ : x t 2 t x () (() ) = − µ 2 Ct t Ex t t E xt E x t Rt t t xx x xx x [ ( () )] ] ] =− = = µµ 22 2 2 4. Auto correlation : gxt xt xt xt ) ()() 1 2 = R t t Ext xt xxpx x dxdx xx [()() ] ( , ) 1 2 1 2 −∞ −∞ 5. Auto covariance of : ( ) and ( ) t t ) {() } {() } 1 1 2 2 = µ µ C t t E x x ttR t xx x x xx x x ( , )[ { ( )( ) } { ( ) } ] ( , ) ( 1 1 2 2 1 2 t ) = µ µ µ µ 6.

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07Stochastic - MAE 591 RANDOM DATA Stochastic Processes A...

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