Econ4261_HWs

Econ4261_HWs - ECON 4261: Introduction to Econometrics Fall...

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Fall 2009 Problem Set 1 Due: Oct 6, 2009 Exercise 1 Let A be an ( m × n ) matrix, with m > n . Show that the matrix A 0 A is symmetric, and that if A has full rank, A 0 A is positive definite. Exercise 2 Characterize each of the following statements as true or false. If a statement is true, show the proof. If it is false, give a counterexample. Assume that A and B are square matrices. (i) If A and B are non-singular, then AB is non-singular. (ii) ( AB ) 0 = A 0 B 0 . (iii) If AB + BA = 0, then A 2 B 3 = B 3 A 2 . (iv) If A 2 = 0, then A = 0 . (v) If A and B are symmetric, then AB is symmetric. Exercise 3 Let X and Y be two random variables. Consider the following joint density function f ( x,y ) = ± A ( x 2 + y 2 ) for x [0 , 1] and y [0 , 1] 0 otherwise (i) Find A. (ii) Find the marginal density f ( x ) . (iii) Find the conditional density f ( y | x ) . 1
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This note was uploaded on 02/23/2011 for the course ECON 4261 taught by Professor Staff during the Fall '08 term at Minnesota.

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Econ4261_HWs - ECON 4261: Introduction to Econometrics Fall...

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