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RAS0103

# RAS0103 - Problem RAS0103 1 1 Compute the mean value...

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Problem RAS0103 1 1. Compute the mean value, correlation, and covariance functions of the following random pro- cesses x ( u,t ) , u ∈ U ,t ∈ T : (a) x ( u,t ) = a i ( u ) t ( i,i + 1] , i ∈ { 0 , ± 1 , ± 2 ,... } Here T is the real line with x ( u,t ) taking on a diﬀerent value in each unit t -interval. As- sume that { a i ( u ) } is a sequence of identically distributed, uncorrelated random variables, each with mean m and variance σ 2 . (b) x ( u,t ) = N X n =1 a n cos( ω n t + φ n ( u )) where T is the real line and { φ n ( u ) } is a set of independent real random variables, each uniformly distributed on [ - π,π ). (c) x ( u,t ) = t X n =1 a n ( u ) where T is the positive integers, and { a i ( u ) } is a sequence of identically distributed, uncorrelated random variables, each with mean m and variance σ 2 . (d) y ( u,t ) = A ( u ) cos[ t + φ ( u )] where T is the real line, and the random variables A ( u ) and φ ( u ) are independent real random variables, A ( u ) has probability density function with mean m and variance σ 2 , and φ ( u ) has probability density p φ ( u ) ( θ ) = 1 π , for 0 θ π and zero elsewhere. Solution 1. Solution: (a) Since { a i ( u ) } i = -∞ are uncorrelated with mean m and variance σ 2 , then Mean (expected) value : m x ( t ) = E { x ( u,t ) } = E { a i ( u ) } = m for t ( i,i + 1]. Hence m x ( t ) = m x = m for all t ( -∞ , ). Correlation : R x ( t 1 ,t 2 ) = E { x ( u,t 1 ) x * ( u,t 2 ) } = E { a i 1 ( u ) a * i 2 ( u ) } where t 1 ( i 1 ,i 1 + 1] ,t 2 ( i 2 ,i 2 + 1].

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RAS0103 - Problem RAS0103 1 1 Compute the mean value...

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