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Unformatted text preview: Computation of the marginal pdfs Let ( X,Y ) be a continuous random vector with a joint density f X,Y ( x,y ). Then X and Y are continuous random variables. The marginal pdfs are f X ( x ) and f Y ( y ). One computes the marginal pdfs by integrating the other variable out of the joint pdf: f X ( x ) = Z ∞∞ f X,Y ( x,y ) dy and f Y ( y ) = Z ∞∞ f X,Y ( x,y ) dx. Example Let f X,Y ( x,y ) = 4( x + y ) 2 , x ≥ , y ≥ , x + y ≤ 1 . Compute the marginal densities. Random vectors in more than 2 dimensions Let X 1 ,X 2 ,...,X k be random variables. The joint cdf of X 1 ,X 2 ,...,X k is F X 1 ,...,X k ( x 1 ,...,x k ) = P ( X 1 ≤ x 1 ,...,X k ≤ x k ) ,∞ < x 1 ,...,x k < ∞ . The marginal cdfs F X 1 ( x 1 ) ,...,F X k ( x k ) can be computed by letting all the extra variables go to infinity: F X 1 ( x 1 ) = F X 1 ,...,X k ( x 1 , ∞ ,..., ∞ ) = lim x 2 →∞ ,...,x k →∞ F X 1 ,...,X k ( x 1 ,x 2 ,...,x k ) , ......
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This note was uploaded on 09/20/2010 for the course OR&IE 3500 at Cornell.
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