hw2 - ECE514 Random Process Fall 2010 HW2 Due Contact...

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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. 1. Let g 1 ( · ) = F - 1 X 1 ( · ) be the inverse finction of F X 1 ( · ) , the marginal CDF of the K -dimensional CDF F X 1 ,...,X K ( x 1 , . . . , x K ) , i.e., F X 1 ( c ) = lim c 2 →∞ ,...,c
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