prv_lec6 - Probability Random Variables Chapter 6 Hyung Il...

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Probability & Random Variables Chapter 6 Hyung Il Koo [email protected] Department of Electrical Engineering Ajou University, Korea Spring 2016
6.1 The Random Process Concept Random process X ( t, s ) A family or ensemble of time functions when t (time) and s (outcome) are variables. Simply denoted by X ( t ) x ( t ) represents a specific waveform of a random process X ( t ) sample function, ensemble member, or realization X ( t ) = {· · · , x n - 1 ( t ) , x n ( t ) , x n +1 ( t ) , · · · } X i denotes the random variable associated with the process X ( t ) at time t i X i = X ( t i , s ) = X ( t i )
6.1 Classification of Processes Let X = X ( t ) at time t : a random variable t : Discrete Sequence, Continuous Process X : Discrete / Continuous
6.1 Classification of Processes Figure: Continuous Random Process Figure: Discrete Random Process
6.1 Classification of Processes Figure: Continuous Random Sequence Figure: Discrete Random Sequence
6.1 Deterministic vs. Nondeterministic Processes If future values of any sample function cannot be predicted exactly from observed past values, the process is called nondeterministic . A process is called deterministic if future values of any sample function can be predicted from past values. E.g. X ( t ) = A cos( ω 0 t + Θ) , A and Θ are random : deterministic process
6.2 Distribution and density functions Recall that for a fixed t , X ( t ) is a random variable. Cumulative Distribution Functions (CDFs) 1 st order CDF: F X ( x 1 ; t 1 ) = P ( X ( t 1 ) x 1 ) 2 nd order joint CDF: F X ( x 1 , x 2 ; t 1 , t 2 ) = P ( X ( t 1 ) x 1 , X ( t 2 ) x 2 ) n th order joint CDF: F X ( x 1 , · · · , x n ; t 1 , · · · , t n ) = P ( X ( t 1 ) x 1 · · · X ( t n ) x n ) Probability Density Functions (PDFs) 1 st order PDF: f X ( x 1 ; t 1 ) = dF X ( x 1 ; t 1 ) dx 1 2 nd order joint PDF: f X ( x 1 , x 2 ; t 1 , t 2 ) = 2 F X ( x 1 ,x 2 ; t 1 ,t 2 ) ∂x 1 ∂x 2 n th order joint PDF: f X ( x 1 , · · · , x n ; t 1 , · · · , t n ) = n F X ( x 1 , ··· ,x n ; t 1 , ··· ,t n ) ∂x 1 ··· ∂x n
6.2 Statistical Independence Two random processes X ( t ) and Y ( t ) are statistically independent if f X,Y ( x 1 , · · · , x N , y 1 , · · · , y M ; t 1 , · · · , t N , t 1 , · · · , t M ) = f X ( x 1 , · · · , x N ; t 1 , · · · , t N ) f Y ( y 1 , · · · , y M ; t 1 , · · · , t M ) N, M ∈ N , t i , t j ∈ ℜ Loosely speaking, if two processes are not related, they are independent.
6.2 First order stationary processes Stationary = Does not change over time A r.p. is called stationary to order one ( 1 st order stationary) if its 1 st order density function does not change with a shift in time origin.

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