hw1 stats practice

# hw1 stats practice - STAT 409 Homework 1 ( Answers ) Fall...

This preview shows pages 1–3. 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: STAT 409 Homework 1 ( Answers ) Fall 2008 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) & 2 X X ln 1 & & ; x x x f x f ⋅- = = , x > 1, θ > 1. a) Find the maximum likelihood estimator & ˆ of θ . L( θ ) = ( ) ∏ = ⋅- n i i i x x 1 & 2 ln 1 & . ln L( θ ) = ( ) & & = = ⋅- +- n i i n i i x x n 1 1 ln & ln ln 1 & 2 ln . ( ) & =-- = n i i d d x n 1 ln 1 & 2 & & L = 0. ¡ & = + = n i i x n 1 ln 2 1 & ˆ . b) Suppose θ > 2. Find the method of moments estimator & ~ of θ . E(X) = ( ) ( ) ( ) ( ) 2 2 1 & 2 X 2 & 1 & ln 1 &-- =- = ¢ ¢ ∞ ∞ ∞- ⋅ ⋅ ⋅ dx x x x dx x f x . ( ) ( ) 2 2 1 2 & 1 & 1-- = = & = ⋅ x x n n i i . ¡ 1 1 2 & ~-- = x x . 2. Forty-eight measurements are recorded to several decimal places. Each of these 48 numbers is rounded off to the nearest integer. The sum of the original 48 numbers is approximated by the sum of these integers. If we assume that the errors made by rounding off are i.i.d. and have uniform assume that the errors made by rounding off are i....
View Full Document

## This note was uploaded on 10/11/2010 for the course STATS 410 taught by Professor Alexey during the Spring '10 term at University of Illinois at Urbana–Champaign.

### Page1 / 5

hw1 stats practice - STAT 409 Homework 1 ( Answers ) Fall...

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

View Full Document
Ask a homework question - tutors are online