stat425fall10exam1-sln

# stat425fall10exam1-sln - 1 1 A biologist gathers data of...

This preview shows pages 1–2. Sign up to view the full content.

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: 1 1. A biologist gathers data of the form ( x i , y i ), i = 1 ,.., n , where x i is the amount of time since the start of the experiment and y i is the amount of pollen that arrives in pollen trap i . She considers the regression model i i i y x e β = + where the i e s are uncorrelated random variables with mean 0 and variance 2 σ . a) What is the least-squares estimate of β ? i i x y β ˆ ˆ = , ) ˆ ( ˆ i i i y y e- = ∑ =- = n i i i x y RSS 1 2 ) ( β ∑ = = =-- = ∂ ∂ n i i i i x y x RSS 1 ˆ ) ˆ ( 2 β β β β ∑ = =- n i i i i x y x 1 ) ˆ ( β , ∑ ∑ = = =- n i n i i i i x y x 1 1 2 ˆ β ∑ ∑ = = = n i i n i i i x y x 1 2 1 ˆ β b) Show that least square estimate β ˆ is unbiased. β β β = = = = ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = n i i n i i i n i i n i i i n i i n i i i x x x x y E x x y x E E 1 2 1 1 2 1 1 2 1 ) ( ) ( ) ˆ ( . Thus, it is unbiased....
View Full Document

## This note was uploaded on 12/12/2010 for the course STAT 425 taught by Professor Ma,p during the Fall '08 term at University of Illinois, Urbana Champaign.

### Page1 / 4

stat425fall10exam1-sln - 1 1 A biologist gathers data of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online