� Definition of Stochastic Processes � Definition of Stochastic Processes st Order Density and Distribution � 1 1 st Order Density and Distribution � nd Order Density and Distribution � 2 � 2 nd Order Density and Distribution � Conditional Density Conditional Density � � Expected Value and Autocorrelation � Expected Value and Autocorrelation � Auto Auto- -covariance and Variance covariance and Variance � � � Examples of Physical Stochastic Examples of Physical Stochastic Processes Processes G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Given a random experiment ℑ ℑ : (S, F, P), to each Given a random experiment : (S, F, P), to each Meaning of X(t, ξ ξ ) ) Meaning of X(t, ξ we assign time function ξ X(t, ξ ξ ) t ∈ ∈ (0,T) we assign time function X(t, ) t (0,T) ( ) t, ξ ξ vary: family of time � t, vary: family of time ξ � X t , Stochastic process Stochastic process ξ 1 functions functions ξ ∈ ∈ S. S. ξ 3 ξ 2 family of time functions, ξ family of time functions, � � t vary: single time t vary: single time ξ 4 functions functions X(t, , ξ ξ ) ) X(t t ξ vary: random variable ξ t 1 ξ 1 � vary: random variable � Example of a stochastic process. Sample space has four outcomes. ξ fixed: single number � t, t, ξ fixed: single number � ξ 3 ξ 2 ( ) ( ) Note: ... are n Note: ... are n X = ξ X = ξ , , X t X t 1 1 n n random variables random variables ξ 4 t 1 t G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 1
Derivative of Air Temp Derivative of Air Temp Over Ocean Derivative of Air Velocity Over Ocean Derivative of Air Velocity Alternative definition : Stochastic process is Alternative definition : Stochastic process is a family of random variables X(t 1 ), X(t 2 ),… a family of random variables X(t 1 ), X(t 2 ),… ∈ (0,T). for t ∈ (0,T). for t i. Discrete Discrete – – set is finite/ set is finite/countably countably infinite infinite i. ii. Continuous ii. Continuous – – set is non set is non- -countably countably infinite infinite G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics st Order Density Function Stratnovich 1 Stratnovich 1 st Order Density Function st Order Density & Distribution Functions 1 st 1 Order Density & Distribution Functions ( ) { ( ( ) ) } = δ − f x ; t E X t x ( ) ( ) { ( ) } ∂ = ≤ ; ( ) F x t , = F x t P X t x ; f x t ∂ x nd Order Density Function Stratnovich 2 Stratnovich 2 nd Order Density Function nd Order Density & Distribution Functions 2 nd Order Density & Distribution Functions 2 ( ) { ( ( ) ) ( ( ) ) } = δ − δ − , ; , f x x t t E X t x X t x 1 2 1 2 1 1 2 2 ( ) { ( ) ( ) } = ≤ ∩ ≤ F x , x ; t , t P X t x X t x 1 2 1 2 1 1 2 2 ( ) ( ) Properties Properties ∞ = F x , ; t , t F x ; t ( ) ∂ 1 1 2 1 1 2 , ; , ( ) F x x t t = 1 2 1 2 , ; , f x x t t 1 2 1 2 ∂ ∂ ( ) +∞ ( ) x x ∫ = ; , ; , 1 2 f x t f x x t t dx 1 1 1 2 1 2 2 − ∞ G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 2
( ) Conditional Density Conditional Density , ; , ( ( ) ) f x x t t = = 1 2 1 2 f x ; t | X t x ( ) 1 2 2 , f x t 2 2 Mean Value Mean Value ( ) { ( ) } +∞ ( ) ∫ η = , ξ = ; t E X t xf x t dx − ∞ Autocorrelation Autocorrelation ( ) { ( ) ( ) } +∞ ( ) ∫ = = , , ; , R t t E X t X t x x f x x t t dx dx 1 2 1 2 1 2 1 2 1 2 1 2 − ∞ ( ) { [ ( ) ( ) ] [ ( ) ( ) ] } Autocovariance Autocovariance = − η − η C t 1 , t E X t t X t t 2 1 1 2 2 Sample time & space variations of fluctuation Sample time & space variations of fluctuation or or ( ) ( ) ( ) ( ) = − η η , , C t t R t t t t velocity components in turbulent near wall flows velocity components in turbulent near wall flows 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) Variance Variance σ = = − η 2 2 t C t , t R t , t t X t G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics STS- -41 Z Lift 41 Z Lift- -Off Acceleration Off Acceleration STS Actual Simulated Actual Simulated Original Original Simulated Simulated Accelerograms of 1940 El Centro Earthquakes Accelerograms of 1940 El Centro Earthquakes G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 3
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