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Unformatted text preview: Homework Assignment 1 STAT 443 Forecasting: Due Date: January 28th 2010 Location: Drop Box: 3rd floor, MC building 1) (10 points) Prove the following statement There exists a linear relationship between two random variables U and V, i.e. V = a+bU, if and only if the correlation coefficient (U,V) has an absolute value of 1 2) (10 points)Show that, if U and V are random variables following a joint Normal distribution, the assumption of independence is equivalent to the assumption of uncorrelatedness. 3) (7 points)Consider the following time series model: Yt= 0 +1Xt +t, t=1.2...... Given historical data upto n periods, find the 95% Prediction interval for Yt, where t>n 4) (5 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); b) Var(X); c) E(2X+3). 5) (10 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? b) Does E(1/X) exist? c) For what values of k does E(X^k) exist? 6) (30 points) In order to manufacture bats for the game of professional cricket, each bat has to be of a specific weight. Every machine that produces cricket bats has a certain target value 0 (representing the ideal weight) and that is aimed by calibrating the machine carefully. The weight of the bats in 30 consecutive days is obtained and is given in the data set below. If the machine is calibrated correctly, and if the OLS assumptions hold, we would expect a constant mean model with the Yt's fluctuating randomly around the target value 0. The data set Yt; t=1,2,...30 37, 27, 25, 32, 29, 30, 31, 33, 34, 38, 33, 31, 37, 41, 26, 32, 37, 31, 25, 35, 36, 39, 31, 32, 28, 34, 30, 27, 26, 38. (i) Find the OLS estimate for 0, i.e 0* , and a 95% confidence interval for 0 (ii) Find a 95% prediction interval for Yt, where t>30 (iii) Run a residual diagnostic test for the OLS assumptions, checking for heteroscedasticity, uncorrelatedness and normality ...
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