hw2b - 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. 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 K →∞ F X 1 ,...,X K ( c,c 2 ,...,c K ) = F X 1 ,...,X K ( c, + ∞ ,..., + ∞ ) . 2. Compute U 1 ∼ U [0 , 1] and x 1 = g 1 ( U 1 ) . 3. Initialize k = 2 . 4. Define g k ( · ) = F- 1 X k | X 1 ,...,X k- 1 ( ·| X 1 = x 1 ,...,X k- 1 = x k- 1 ) as the inverse finction of the marginal CDF of X k conditioned on all previous values ( X 1 = x 1 ,...,X k- 1 = x k- 1 ) that have been generated, F X k | X 1 ,...,X k- 1 ( X k ≤ c k | X 1 = x 1 ,...,X k- 1 = x k- 1 ) = R c u k =-∞ f X 1 ,...,X k ( X 1 = x 1 ,...,X k- 1 = x k- 1 ,X k = u k ) du k R ∞ u k =-∞ f X 1 ,...,X k ( X 1 = x 1 ,...,X k- 1 = x k- 1 ,X k = u k ) du k (1) 5. Compute U k ∼ U [0 , 1] and x k = g k ( U k ) . 6. If k = K then halt; else go to Step 4. We now prove that this construction generates ( X 1 ,...,X k ) with the correct CDF. Our proof is via induction. (a) Prove the basis of the induction, that is, prove that Step 2 generates X 1 whose marginal distribution follows that of the K-dimensional CDF. Hint: this part is straightforward from previous material. Answer : In the review material on probability (Chapter 1), we saw that in order to generate a random variable (RV) with cumulative distribution function (CDF) F X ( · ) , it suffices to first generate a uniformly distributed RV U ∼ [0 , 1] and map it to X via the inverse CDF. More formally, F X ( · ) : R → [0 , 1] , F- 1 X ( · ) : U [0 , 1] → R , X ∼ F X ( · ) , U ∼ U [0 , 1] , u = F X ( x ) , x = F- 1 X ( u ) , and F- 1 X ( F X ( x )) = x ....
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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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hw2b - ECE514 Random Process Fall 2010 HW2 - Due: September...

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