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ps10 - ECSE-305 Winter 2009 Probability and Random Signals...

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ECSE-305, Winter 2009 Probability and Random Signals I Assignment #10 Posted: Thursday, April 2, 2009. Due: Tuesday, April 14, 2009, 11h00am, MC756. Notes: Assignments without this cover page will be discarded. Student #1: Name: ID: Student #2: Name: ID: Question Marks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Total
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1. Let X 1 , X 2 , . . . be independent and identically distributed (i.i.d.) ran- dom variables with P ( X n = 1) = p and P ( X n = - 1) = q = 1 - p , for all n . Define Y n = n summationdisplay i =1 X i n = 1 , 2 , . . . and Y 0 = 0. The collection of RVs { Y n : n 0 } is a random process, called a random walk . (a) What type of process is Y n ? Identify the index parameter T and the state space Ω. (b) Construct a typical realization of Y n in the case p = 1 / 2. (Hint: use a coin...) (c) Find the mean and variance function, i.e. μ Y ( n ) and σ 2 Y ( n ), of the process Y n . 2. Consider a random process X ( t ) defined by X ( t ) = A sin(2 πFt ) , t 0 where the amplitude A is a discrete RV with P ( A = 1) = P ( A = - 1) = 1 / 2, and the frequency F is a discrete RV with P ( F = 1) = P ( F = 2) = 1 / 2.
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