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SolSet1 - EE 562a Homework Solutions 1 September 10, 2009 1...

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Unformatted text preview: EE 562a Homework Solutions 1 September 10, 2009 1 1. Solution: The cummulative distribution function can be evaluated using conditional probabilities. P x ( u,t ) ( z ) = P { u : x ( u,t ) z } = P { u = u } P { x ( u,t ) z | u = u } + P { u = u 1 } P { x ( u,t ) z | u = u 1 } = 1 2 P { 2sin( t ) z } + 1 2 P { 2 t z } Since the calculation has been conditioned down to single points in the sample space, the probabilities in this last expression are either 1 or 0 depending on whether the event is is true or false (a function of the choice of random variable index t ). Hence P x ( u,t ) ( z ) = 1 2 U r ( z- 2sin( t )) + 1 2 U r ( z- 2 t ) . Here U r ( ) denotes the right-continuous unit-step function, U r ( z ) = , z < 1 , z . In the above computation we first considered calculations for a particular u { u ,u 1 } by conditioning on different sample space points being given. Lets try the computation a different way by first fixing t . For example, consider the case in which t = 0. Then { u : x ( u, 0) z } = , z < { u ,u 1 } , z where denotes the empty set. Therefore P x ( u,t ) ( z ) = P { u : x ( u,t ) z } = P { } , z < P { u ,u 1 } , z = , z < 1 , z = U r ( z ) The following special cases special plotted in Figure 1: P x ( u, 0) ( z ) = U r ( z ) P x ( u, 1 4 ) ( z ) = 1 2 [ U r ( z- 2) + U r ( z- 1 2 )] P x ( u, 1 2 ) ( z ) = 1 2 [ U r ( z- 2) + U r ( z- 1)] P x ( u, 1) ( z ) = 1 2 [ U r ( z ) + U r ( z- 2)] 2. This problem challenges the student to use distribution functions in the analysis of simple transformations of random variables. (a) The student should be able to prove that when the strictly monotone-increasing function g ( z ) performs a one-to-one transformation of the real line into itself. Hence every value of w = g ( z ) can be uniquely associated with the value of z that created it through an inverse function z = g- 1 ( w ). (Dont confuse this with the reciprocal [ g ( w )]- 1 .) Generally P y ( u ) ( z ) , P { u : y ( u ) z } = P { u : g ( x ( u )) z } 2 EE 562a Homework Solutions 1 September 10, 2009 1 1 P x ( u, 0) ( z ) P x ( u, . 25) ( z ) 2 2 1 / 4 2 1 1 P x ( u, . 5) ( z ) P x ( u, 1) ( z ) 1 2 2 Figure 1: Special case plots Using the strictly monotone-increasing property reduces this to P y ( u ) ( z ) = P { u : x ( u ) g- 1 ( z ) } = P x ( u ) ( g- 1 ( z )) . This result holds for all possible values that y ( u ) can take on. We note that g L , lim z - g ( z ) y ( u ) lim z g ( z ) , g U , and hence the complete answer is P y ( u ) ( z ) = , z < g L P x ( u ) ( g- 1 ( z )) , g L z g U 1 , g U < z. (b) In this case some of the inequalities are reversed and that causes minor changes. Using the strictly monotone-decreasing property on the first equation in (a) gives the result P y ( u ) ( z ) = P { u : x ( u ) g- 1 ( z ) } = 1- P { u : x ( u ) < g- 1 ( z ) } = 1- lim z z P x ( u ) ( g- 1 ( z...
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SolSet1 - EE 562a Homework Solutions 1 September 10, 2009 1...

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