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ps1[1] - Problem Set 1 ECON 837 Prof Simon Woodcock Spring...

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Problem Set 1 ECON 837 Prof. Simon Woodcock, Spring 2006 Due: Friday, Jan. 27 in class (Some problems from Casella and Berger Statistical Inference , 1990). 1. Prove that the following functions are cdfs: (a) 1 2 + 1 ° tan ° 1 ( x ) ; x 2 ( °1 ; 1 ) (b) (1 + e ° x ) ° 1 ; x 2 ( °1 ; 1 ) (c) e ° e ° x ; x 2 ( °1 ; 1 ) (d) 1 ° e ° x ; x 2 (0 ; 1 ) 2. Let X be a continuous random variable with pdf f ( x ) and cdf F ( x ) : For a °xed number x 0 ; de°ne the function g ( x ) = ° f ( x ) = [1 ° F ( x 0 )] x ± x 0 0 x < x 0 : Prove that g ( x ) is a pdf. (Assume that F ( x 0 ) < 1 ). 3. Suppose that F Y ( y ) = 1 ° y ° 2 for 1 ² y < 1 : (a) Verify that F Y ( y ) is a cdf. (b) Find the pdf of y; f Y ( y ) : (c) Let Z = 10 ( Y ° 1) : Find F Z ( z ) : 4. In each of the following, °nd the pdf of Y and show that the pdf integrates to 1. (a) f X ( x ) = 1 2 e °j x j for x 2 R and Y = j X j 3 (b) f X ( x ) = 3 8 ( x + 1) 2 for ° 1 < x < 1 and Y = 1 ° X 2 (c) f X ( x ) = 3 8 ( x + 1) 2 for ° 1 < x < 1 and Y = 1 ° X 2 if X ² 0 ; and Y = 1 ° X if X > 0 : 5. Prove the following theorem. Let X have continuous cdf F X ( x ) and de°ne the random variable Y = F X ( X ) : Then Y is uniformly distributed on (0 ; 1) : That is, Pr ( Y ²
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