Lecture 11_notes.pdf - Econ 41(Summer 2017 Department of...

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Econ 41 ( Summer 201 7 ) Department of Economics, UCLA Instructor: Ming Gu Lecture 11: Normal Distribution Remark 1 Skip Example 3.4-7 De°nition 2 The random variable X has a normal distribution if its PDF is given by f ( x ) = 1 b p 2 ° exp " ° ( x ° a ) 2 2 b 2 # ; °1 < x < 1 for some a and b satisfying °1 < a < 1 and b > 0 . Remark 3 exp [ t ] means e t . Theorem 4 Z 1 °1 1 b p 2 ° exp " ° ( x ° a ) 2 2 b 2 # dx = 1 ; Z 1 °1 x b p 2 ° exp " ° ( x ° a ) 2 2 b 2 # dx = a; Z 1 °1 ( x ° a ) 2 b p 2 ° exp " ° ( x ° a ) 2 2 b 2 # dx = b 2 : De°nition 5 We usually write f ( x ) = 1 ± p 2 ° exp " ° ( x ° ² ) 2 2 ± 2 # ; °1 < x < 1 and say that the random variable X with this PDF is N ( ²; ± 2 ) . This is compactly written X ± N ( ²; ± 2 ) . De°nition 6 If Z ± N (0 ; 1) , we say that Z has a standard normal distribution. Obviously the PDF of N (0 ; 1) is f ( x ) = 1 p 2 ° exp ° ° x 2 2 ± ; °1 < x < 1 : 1
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De°nition 7 Suppose that Z ± N (0 ; 1) . We write ° ( z ) = Pr [ Z ² z ] = Z z °1 1 p 2 ° exp ° ° x 2 2 ± dx: Theorem 8 It is not possible to evaluate ° ( z ) = Pr [ Z ² z ] = Z z °1 1 p 2 ° exp ° ° x 2 2 ± dx by °nding an antiderivative that can be expressed as an elementary function. We need to use
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