Homework 1

# Homework 1 - Homework 1 STAT 443 Spring 2009 1. (15 points)...

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Homework 1 STAT 443 Spring 2009 1. (15 points) Let X be a discrete random variable that attains values 1, 2 and 5 with probability 1/8, 1/4 and 5/8 respectively. Find: a) E(X); E(X) =1*P(X=1) + 2*P(X=2) + 5*P(X=5)= =1*1/8 + 2*1/4 + 5*5/8= =1/8 + 4/8 + 25/8= =30/8 = 15/4= 3.75 b) Var(X)=E(X^2)- [E(X)]^2=1*1/8 + 4*1/4 + 25*5/8 - 225/16= =1/8+ 1 + 125/8 - 225/16= =1 + 126/8 - 225/16= =1 + 252/16 - 225/16 = =1 + 27/16 = 43/16 = 2.6875 c) E(2X+3)=2*E(X)+3 = 15/2 + 6/2= 21/2 2. (15 points) Let X be a continuous random variable with probability density function f(x) =1/x^2 if 1<x< and zero otherwise. a) Does E(X) exist? No E(X)= ( 29 = = dx x dx x x ' 1 1 2 ln 1 * b) Does E(1/X) exist? Yes. E(1/X)= 2 1 1 * 2 1 1 * 1 1 2 1 3 1 2 = - = = - x xdx x dx x x c) For what values of k does E(X^k) exist? E(X^k) exists for k<1 E(X 2 )= = = dx dx x x 1 1 2 2 1 1 * E( ) k X = - - - = = 1 1 1 2 1 2 1 1 * k X dx X dx x x k k k , for k 1 3. (30 points)1 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 1960 to 1995. 1

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VARIABLE DESCRIPTIONS: YEAR -- from 1960 to 1995 (the training set is from 1960 to 1993) CONS –- per capita real consumption TBILL –- 3 month Treasure bill rate RSERV -- services, in billion dollars of 1992. Values are aligned and delimited by blanks. YOUR TASK: To forecast per capita consumption for 1994 and 1995 using the 3 month Treasury Bills rates and services. The training window is from 1960-1993. You should a) construct a linear regression of per capita consumption vs. Treasury bills rates and services, include the summary output of the linear regression, the scatterplot, correlation, covariance and R-squared of CONS vs. TBILL and RSERV; F igure 1 2 4 6 8 10 12 14 8000 12000 Plot of Cons vs Tbill Treasure bill rate Per capita real consumption 1000 1500 2000 2500 Plot of Cons vs Rserv Services From the second plot (Plot of Cons vs. Rserv ), Figure 1, we observed that there is a positive linear relationship between the response Y and the independent variable Services. In the first plot 2
(Plot of Cons vs.Tbill ), Figure 1, the relationship between Y and the independent variable Treasure bill rate look at the beginning linear, but can we consider even the quadratic shape. data<-read.table("C:/Documents and Settings/Administrator.RETESTRAK/Desktop/12.txt", header=TRUE) attach(data) newdata=data cons<-newdata\$CONS tbill<-newdata\$TBILL rserv<-newdata\$RSERV par(mfrow=c(2,1)) plot(tbill, cons, main="Plot of Cons vs Tbill", xlab="Treasure bill rate", ylab="Per capita real consumption") plot(rserv, cons, main="Plot of Cons vs Rserv", xlab="Services", ylab="Per capita real consumption") Call: lm(formula = cons ~ tbill + rserv)

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## This note was uploaded on 06/21/2009 for the course STAT 443 taught by Professor Yuliagel during the Spring '09 term at Waterloo.

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Homework 1 - Homework 1 STAT 443 Spring 2009 1. (15 points)...

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