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##### Let X and Y have a joint density given by

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Question 1;Let X and Y have a joint density given by;f (X, Y) = a(1 X);0 X 1;0Y 1;and zero elsewhere. Where, k is a constant.;(a) Find the value of a that makes this a probability density function. Use this;value for a in the pdf to work out the remaining questions.;(b) Find the marginal distribution of X and Y, i.e. f (X) and f (Y).;(c) Find E (X), E (Y), E (Y |X) and V ar(X |Y).;(d) Find E [6X 12Y ] and V ar(2X 3Y);(e) Find Cov (XY) and the correlation xy;(f) Are X and Y independent? Show this using two alternative ways.;Question 2;(a) We showed (in class) that the formula used to estimate the slope coecient in;a simple linear regression model, Yt = 1 + 2 Xt + et is given by;2 =;T;Xt Yt Xt Yt;2;T;Xt (Xt)2;Prove this is equivalent to;2 =;(Xt X)(Yt Y);(Xt X)2;(b) We have also shown that, given any two random variables, X and Y;V ar(aX + bY) = a2 V ar(X) + b2 V ar(Y) + 2abCov (X, Y);where a and b are constants. Now suppose you have another random variable Z;and constant c;(i) What would the following simplify to?;V ar(aX + bY + cZ);(ii) What is X, Y and Z were independent, would the result in (i) change?;3;Question 3;Consider the following ve observations on Yt = {5, 2, 3, 2, 2} and Xt = {3, 2, 1, 1, 0};(a) Find 1 and 2;(b) On a graph, plot the data points and sketch the tted regression line, Yt.;Interpret your regression.;(c) On the Sketch in part (b), locate the point of the means, (X, Y). Does the;tted line pass through that point?;(d) Calculate et and V ar(2);(e) let;wt =;Xt X;(Xt X)2;Calculate the following;(i);(ii);wt;wt Xt;Question 4;Two fair coins are each tossed two times. Let X and Y denote the number of heads;showing on the rst and second coins, respectively, after two tosses. Let U = X + Y;and V = X Y.;(a) Find the mean and the variance of V.;(b) Find the covariance of U and V.;Question 5;Given the following data for two independent discrete random variables;X:= {1, 3, 6, 2, 4} and Y:= {5, 2, 7, 9};Evaluate the following;(a);5;2;j =1 (Yj;(b);4;i=1;4;2;j =1 (2Xi;(c);2;i=1;3;j =1 (Xi;2Yj)2;+ 4Yj + 3Xi Yj);X)2 (Yj Y)

Paper#31554 | Written in 18-Jul-2015

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