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HW2 - X T,T = R are i.i.d with X 1 ∼ L(1(a Choose a...

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ECE4110 Homework 2 Fall 09 Cornell University T.L. Fine This assignment is to be handed in at the end of class on Thursday, 10 September. 1. A random process X T , T = N , has mutually independent random vari- ables with common cdf F satisfying ( x R ) 1 > F ( x ) > 0 . (a) If F μ is the cdf for μ = sup t T X t , show that ( x R ) F μ ( x ) = 0 . (b) Evaluate lim x →∞ F μ ( x ). (c) Is there something odd about the answer to (b)? 2. Assume the hypotheses at the outset of Problem 1. (a) If we take a boundary function b t = 0 for all t T , what is the probability of infinitely many upcrossings? (b) What is the probability of only finitely many upcrossings? 3. The random variables in
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Unformatted text preview: X T ,T = R , are i.i.d. with X 1 ∼ L (1). (a) Choose a boundary function b t that is everywhere finite and the proba-bility is one of finitely many upcrossings. (b) Repeat (a) with the change that you are asking about infinitely many upcrossings. 4. Consider the random process X t = Y cos( t ) + Z sin( t ) for T = R ,Y ∼ N (0 , 1) ,Z ∼ E (2) . (a) Evaluate the expected value EX t . (b) If the covariance COV ( Y,Z ) = 0, evaluate the correlation function R X ( t,s ) = E ( X t X s ). (c) Under the above assumptions, evaluate EX 2 t and V AR ( X t ). 1...
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