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
Unformatted text preview: Homework 1 Solution Stat 443 Spring 2007 1. ( 15 points ) Let X be a discrete random variable that attains values 1, 2 and 4 with probability 1/6, 1/3 and 1/2 respectively. Find: a) E ( X ); b) V ar ( X ); c) E (4 X + 5). Solution: a) E ( X ) = 1 1 / 6 + 2 1 / 3 + 4 1 / 2 = 17 / 6; b) V ar ( X ) = 1 / 6 (1 17 / 6) 2 + 1 / 3 (2 17 / 6) 2 + 1 / 2 (4 17 / 6) 2 = 53 / 36; c) E (4 X + 5) = 4 E ( x ) + 5 = 4 17 / 6 + 5 = 49 / 3. 2. ( 15 points )Let X be a continuous random variable with probability density func tion f ( x ) = 1 /x 2 if 1 < x < and zero otherwise. a) Does E ( X ) exist? b) Does E (1 /X ) exist? c) For what values of k does E ( X k ) exist? Solution: a) E ( X ) does not exist because E ( x ) = Z 1 x 1 x 2 dx = ln ( x )  1 = b) E (1 /X ) exists since E (1 /x ) = Z 1 1 x 1 x 2 dx = 1 / 2 x 2  1 = 1 / 2 c) If k < 1 E ( X k ) = Z 1 x k 1 x 2 dx = 1 k 1 x k 1  1 = 1 1 k if k > 1 E ( X k ) = Z 1 x k 1 x 2 dx = 1 k 1 x k 1  1 = if k = 1 E ( x ) = from b). So, E ( X k ) exists when k < 1. 1 2. ( 30 points ) DESCRIPTIVE ABSTRACT: The dataset consumpt.txt contains yearly data on 3 month Treasury bill rate (TBILL) in US dollars and per capita real consumption (CONS) in USA from 1964 to 1995. VARIABLE DESCRIPTIONS: YEAR from 1964 to 1995 (the training set is from 1964 to 1992) CONS per capita real consumption TBILL  3 month Treasure bill rate Values are aligned and delimited by blanks. YOUR TASK: To forecast per capita consumption for 1993, 1994 and 1995 using the 3 month Treasury Bills rates. The training window is from 19641992. You should a) construct a simple linear regression of per capita consumption and Treasury bills rates, include the summary output of the linear regression, the scatterplot,...
View
Full
Document
This note was uploaded on 01/12/2012 for the course STAT 443 taught by Professor Yuliagel during the Spring '09 term at Waterloo.
 Spring '09
 YuliaGel
 Forecasting, Probability

Click to edit the document details