zz-29 - Utah State University ECE 6010 Stochastic Processes...

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ECE 6010 Stochastic Processes Homework # 7 Solutions 1. Suppose { X t , t 0 } is a homogeneous Poisson process with parameter λ . Defne a random vari- able τ as the time oF the frst occurrence oF an event. ±ind the p.d.F. and the mean oF τ . p.d.f. : We have, P ( X t = k ) = e - λt ( λt ) k k ! So, P ( τ > t ) = P ( X t = 0) = e - λt and P ( τ t ) = 1 - e - λt , i.e. F τ ( t ) = 1 - e - λt . ThereFore, f τ ( t ) = λe - λt Mean : E [ τ ] = Z 0 τλe - λτ = - τe - λτ - 1 λ e - λτ ± ± ± ± 0 = - 0 - 0 + 0 + 1 λ = 1 λ 2. Suppose { X t , t R } is a w.s.s. random process with autocorrelation Function R X ( τ ). Show that iF R X is continuous at τ = 0 then it is continuous For all τ R . [ R X ( τ + δ ) - R X ( τ )] 2 = ( E [ X τ + δ X 0 ] - E [ X τ X 0 ]) 2 = ( E [ X 0 ( X τ + δ - X τ )]) 2 E [ X 2 0 ] E [( X τ + δ - X τ ) 2 ] (Cauchy-Schwartz) = R X (0)[ R X (0) - 2 R X ( δ ) + R X (0)] = 2 R X (0)[ R X (0) - R X ( δ )] = 2 R X (0) ε So, ⇒ | R X ( τ + δ ) - R X ( τ ) | ≤ p 2 R X (0) ε ⇒ | R X ( τ + δ ) - R X ( τ ) | < ε 0 So continuous at all τ . 3. Under the conditions oF problem 2, show that For a > 0, P ( | X t + τ - X t | ≥ a ) 2( R X (0) - R X ( τ )) a 2 . Let g ( x ) = x 2 . g is non-negative, non-decreasing on [0 , ) and symmetric about 0. Then, P ( | X | ≥ b ) E [ g ( x )] b 2 . 1
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zz-29 - Utah State University ECE 6010 Stochastic Processes...

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