stat 443 - Homework 1 Solution Stat 443 Spring 2007 1. ( 15...

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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 1964-1992. 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,...
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This note was uploaded on 01/12/2012 for the course STAT 443 taught by Professor Yuliagel during the Spring '09 term at Waterloo.

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stat 443 - Homework 1 Solution Stat 443 Spring 2007 1. ( 15...

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