st422%20hw03%20solution

st422%20hw03%20solution - Solutions: Homework 3: 3 1 2 1 2...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions: Homework 3: 3 1 2 1 2 1 2 3 3 1 2 1 8.2 : ) ( ) ( (1 ) ) = aE( ) (1 ) ( ) = (a+1-a) = b) Given and are independent, to choose 'a' that would minimize Var( ) ( ) ( (1 ) ) , a E E a a a E V V a a = + - + - = + - 2 1 2 2 2 3 1 2 2 2 2 2 1 2 3 2 2 1 2 2 2 2 2 1 2 are independent .Thus, cov(a ,(1 ) ) ( ) ( ) (1 ) ( ) = a (1 ) ( ) , 2 2(1 ) Also, to check if this is the minimum value of 'a', d a V a V a V a dV Thus da a a a - = = + - + - = -- = = + 2 2 2 3 1 2 2 2 2 3 2 2 1 2 ( ) 2( ) 0, da a= minimizes V( ) V = + > + 1 1 1 1 1 2 1 2 2 1 2 3 1 2 3 1 2 4 1 2 3 (1) (1 8.4 ) Given Y exp( ), to find estimators which are unbiased: ( ) ( ) ( ) / 2 ( ) [ ( ) ( )]/ 2 2 / ( 2 ) /3 ( ) [ ( ) 2 ( )]/3 ( 2 ) / min( , , ) ( a Y E E Y Y Y E E Y E Y Y Y E E Y E Y Y Y Y Y f y = = = = + = + = = = + = + = + = = = (1) 1 ) 4 4 5 4 1 ) ( )(1 ( ) Here n=3 f(y) = (1/ )exp(-y/ ) y>0 = 0 elsewhere F(y)=P(Y y)=1-exp(-y/ ) y>0 (y) = (3/ )exp(-3y/ ) exp( /3) E( )= /3 ( ) (1/ ) ( ) / , n y n i i nf y F y f Y E n E Y n n Thus...
View Full Document

Page1 / 8

st422%20hw03%20solution - Solutions: Homework 3: 3 1 2 1 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online