# zz-17 - A and B 5 Let p k t s = X t-X s where X t is a...

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Utah State University ECE 6010 Stochastic Processes Homework # 8 Due Friday October 31, 2003 1. Suppose { X t , t 0 } is a homogeneous Poisson process with parameter λ . Defne a random variable τ as the time o± the frst occurrence o± an event. Find the p.d.±. and the mean o± τ . 2. Suppose { X t , t R } is a w.s.s. random process with autocorrelation ±unction R X ( τ ). Show that i± R X is continuous at τ = 0 then it is continuous ±or all τ R . (Hint: Use the Schwartz inequality.) 3. Under the conditions o± problem 2, show that ±or a > 0, P ( | X t + τ - X t | ≥ a ) 2( R X (0) - R x ( τ )) a 2 . 4. Suppose A and B are random variables with E [ A 2 ] < and E [ B 2 ] < . Defne the random processes { X t , t R } and { Y t , t R } by X t = A + Bt Y t = B + At, t R . Find the mean, autocorrelation, and cross correlations o± these random processes in terms o± the moments o±

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Unformatted text preview: A and B . 5. Let p k ( t, s ) = X t-X s , where X t is a homoegenous Poisson counting process with rate Î» . Show that the diÂ²erential equation âˆ‚ âˆ‚t p k ( t, s ) = Î» [ p k-1 ( t, s )-p k ( t, s )] is solved by p k ( t, s ) = e-Î» ( t-s ) ( Î» ( t-s )) k k ! k = 0 , 1 , . . . , t > s â‰¥ . 1 6. Let p k ( t, s ) = X t-X s , where X t is an inhomoegenous Poisson counting process with time-varyng rate Î» t . Show that the diferential equation âˆ‚ âˆ‚t p k ( t, s ) = Î» t [ p k-1 ( t, s )-p k ( t, s )] is solved by p k ( t, s ) = e-Î» R t s Î» x dx ( R t s Î» x dx ) k k ! k = 0 , 1 , . . . , t > s â‰¥ . 2...
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zz-17 - A and B 5 Let p k t s = X t-X s where X t is a...

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