Lecture12-2010 - Normal Random Variables Charles B. Moss...

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Normal Random Variables Charles B. Moss July 22, 2010 I. Univariate Normal Distribution. A. Defnition 5.2.1. The normal density is given by f ( x )= 1 σ 2 π exp " 1 2 ( x μ ) 2 σ 2 # −∞ <x< ,σ> 0( 1 ) B. Theorem 5.2.1 Let X be N ( μ, σ 2 ) as defned in Defnition 5.2.1, then E [ X ]= μ and V [ X σ 2 . 1. Starting with the defnition oF the expectation E[ X Z −∞ 1 σ 2 π x exp " 1 2 ( x μ ) 2 σ 2 # dx (2) Using the change in variables technique, we create a new ran- dom variable z such that z = x μ σ x = + μ dx = σdz (3) Substituting into the original integral yields X Z −∞ 1 σ 2 π ( + μ )exp ± 1 2 z 2 ² dz = Z −∞ 1 σ 2 π exp ± 1 2 z 2 ² + μ Z −∞ 1 2 π exp ± 1 2 z 2 ² dz (4) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XII Fall 2010 Taking the integral of the ±rst term ±rst, we have Z −∞ 1 σ 2 π 2 exp ± 1 2 z 2 ² dx = C Z −∞ z exp ± 1 2 z 2 ² dz = C exp ± 1 2 z 2 ² ⎢
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Lecture12-2010 - Normal Random Variables Charles B. Moss...

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