# 09_16 - x , 0 &amp;amp;lt; x &amp;amp;lt; , zero elsewhere....

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

STAT 410 Examples for 09/16/2011 Fall 2011 Let X 1 and X 2 have joint p.d.f. f ( x 1 , x 2 ) . S = { ( x 1 , x 2 ) : f ( x 1 , x 2 ) > 0 } – support of ( X 1 , X 2 ) . F ( x 1 , x 2 ) = P ( X 1 x 1 , X 2 x 2 ) . f ( x 1 , x 2 ) = 2 F ( x 1 , x 2 ) / x 1 x 2 . Let Y 1 = u 1 ( X 1 , X 2 ) and Y 2 = u 2 ( X 1 , X 2 ) . y 1 = u 1 ( x 1 , x 2 ) y 2 = u 2 ( x 1 , x 2 ) one-to-one transformation maps S onto T – support of ( Y 1 , Y 2 ) . x 1 = w 1 ( y 1 , y 2 ) x 2 = w 2 ( y 1 , y 2 ) J = 2 2 1 2 2 1 1 1 y x y x y x y x The joint p.d.f. g ( y 1 , y 2 ) of ( Y 1 , Y 2 ) is g ( y 1 , y 2 ) = f ( w 1 ( y 1 , y 2 ) , w 2 ( y 1 , y 2 ) ) | J | ( y 1 , y 2 ) T 0 elsewhere.

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

View Full Document
1. Let X 1 and X 2 have joint p.d.f. f ( x 1 , x 2 ) = 2 e ( x 1 + x 2 ) , 0 < x 1 < x 2 . a) Find the joint p.d.f. g ( y 1 , y 2 ) of the variables Y 1 = X 2 – X 1 and Y 2 = X 1 . b) Find the joint p.d.f. h ( z 1 , z 2 ) of the variables Z 1 = X 1 + X 2 and Z 2 = X 2 / X 1 .
2. Let X and Y have the joint probability density function f X, Y ( x , y ) = < + > > otherwise 0 1 , 0 , 0 60 2 y x y x y x Let U = X Y and V = X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 3. 2.7.1 Let X 1 , X 2 , X 3 be iid, each with the distribution having pdf f ( x ) = e

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: x , 0 &lt; x &lt; , zero elsewhere. Show that Y 1 = 2 1 1 X X X + , Y 2 = 3 2 1 2 1 X X X X X + + + , Y 3 = X 1 + X 2 + X 3 are mutually independent. 4. 2.2.5 Let X 1 and X 2 be continuous random variables with the joint probability density function, f X 1 , X 2 ( x 1 , x 2 ), &lt; x i &lt; , i = 1, 2. Let Y 1 = X 1 + X 2 and Y 2 = X 2 . (a) Find the joint pdf f Y 1 , Y 2 . (b) Show that ( ) ( ) --= 2 2 2 1 X , X 1 Y , 2 1 1 dy y y y f y f ( 2.2.1 ) which is sometimes called the convolution formula . 5. 2.2.7 Use the formula ( 2.2.1 ) to find the pdf of Y 1 = X 1 + X 2 , where X 1 and X 2 have the joint pdf f X 1 , X 2 ( x 1 , x 2 ) = 2 e ( x 1 + x 2 ) , 0 &lt; x 1 &lt; x 2 &lt; , zero elsewhere....
View Full Document

## This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

### Page1 / 4

09_16 - x , 0 &amp;amp;lt; x &amp;amp;lt; , zero elsewhere....

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

View Full Document
Ask a homework question - tutors are online