10_27_2ans - STAT 400 . Fall 2011 Examples for 10/27/2011...

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STAT 400 Examples for 10/27/2011 (2) Fall 2011 ¼ . Let Z be a N ( 0, 1 ) standard normal random variable. Then X = Z 2 has a chi-square distribution with 1 degree of freedom. M X ( t ) = E ( e t Z 2 ) = - - z e e d z z t 2 2 2 2 1 π = ( ) ( ) - - - z e d t z 2 2 1 2 2 1 π = ( ) 2 1 2 1 1 t - , t < 1 / 2 , since ( ) ( ) ( ) 2 2 1 2 2 1 2 2 1 π t z e t - - - is the p.d.f. of a N ( 0, t 2 1 1 - ) random variable. X has a χ 2 ( 1 ) distribution. ½ . Let X and Y be be two independent χ 2 random variables with m and n degrees of freedom, respectively. Then W = X + Y has a chi-square distribution with m + n degrees of freedom. M X ( t ) = ( ) 2 2 1 1 m t - , t < 1 / 2 , M Y ( t ) = ( ) 2 2 1 1 n t - , t < 1 / 2 . M W ( t ) = M X ( t ) M Y ( t ) = ( ) ( ) 2 2 1 1 n m t + - , t < 1 / 2 . W has a χ 2 ( m + n ) distribution.
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1. A manufacturer of TV sets wants to find the average selling price of a particular model. A random sample of 25 different stores gives the mean
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This note was uploaded on 11/07/2011 for the course STAT 400 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_27_2ans - STAT 400 . Fall 2011 Examples for 10/27/2011...

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