h6soln

# h6soln - Homework 6 Solutions Enrique Campos-N´ a˜nez The...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework # 6 Solutions Enrique Campos-N´ a˜nez 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 (...
View Full Document

## This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.

### Page1 / 9

h6soln - Homework 6 Solutions Enrique Campos-N´ a˜nez The...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online