hw2 - ECE514 Random Process Fall 2010 HW2 - Due: September...

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Unformatted text preview: ECE514 Random Process Fall 2010 HW2 - Due: September 30, 2010 Contact [email protected] 1. A classical approach to generate an RV X with arbitrary CDF F X ( · ) is to apply the inverse CDF to a uniformly distributed RV, i.e., X = F- 1 X ( U ) , U ∼ [0 , 1] . This approach was outlined in class, appears in Chapter 1, and is related to previous homework and quiz questions. Using this transformation F- 1 : U ∈ [0 , 1] → X ∈ R , we can leverage the progress that has been made in development of random number generators for uniformly distributed outputs. In this question, we will show how to construct a function that maps a single uniformly distributed RV to K dependent RV’s that satisfy an arbitrary CDF. To show this result, we will proceed in several steps. First, we show that K independent uniform RV’s can be mapped to an arbitrary K-RV CDF, i.e., ( X 1 ,...,X K ) = g ( U 1 ,...,U K ) , where U k ∼ U [0 , 1] , k ∈ { 1 ,...,K } , ( U k ) K k =1 are independent, ( X 1 ,...,X K ) ∼ F X 1 ,...,X K ( x 1 ,...,x K ) for arbitrary CDF F ( · ) , and g : [0 , 1] K → R K . Construct ( x 1 ,...,x K ) iteratively as below....
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This note was uploaded on 01/25/2011 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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hw2 - ECE514 Random Process Fall 2010 HW2 - Due: September...

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