h6soln - Homework # 6 Solutions Enrique Campos-N anez The...

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Unformatted text preview: Homework # 6 Solutions Enrique Campos-N anez The George Washington University ApSc 116 1 / 9 Problem (Page 175 # 1, Section 3.9) Suppose that X 1 and X 2 are i.i.d. random variables and that each of them has a uniform distribution on the interval [0 , 1] . Find the p.d.f. of Y = X 1 + X 2 . There are several ways to proceed. 1) Find the d.f. of Y by integration (done in class), 2) use the multivariate transformation result, or 3) use the convolution formula. For 2), we can define y = r 1 ( x 1 , x 2 ) = x 1 + x 2 , and z = r 2 ( x 1 , x 2 ) = x 1 . The reason for defining both is that we need to be able to have an invertible or bijective transformation (recuperate the x i s from y and z-notice that you cannot do this with only y ). Now, given z , y , we can find x 1 = s 1 ( y , z ) = z , and x 2 = s 2 ( y , z ) = y- z . The Jacobian is J = 1 1- 1 , det ( J ) =- 1 2 / 9 Therefore, the joint p.d.f. of Y and Z is g ( y , z ) = f ( s 1 ( y , z ) , s 2 ( y , z )) | J | = 1 , since f (...
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h6soln - Homework # 6 Solutions Enrique Campos-N anez The...

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