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Lecture12-2010

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

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Normal Random Variables Charles B. Moss July 22, 2010 I. Univariate Normal Distribution. A. Definition 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 defined in Definition 5.2.1, then E [ X ] = μ and V [ X ] = σ 2 . 1. Starting with the definition of the expectation E [ X ] = −∞ 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 E [ X ] = −∞ 1 σ 2 π ( + μ ) exp 1 2 z 2 dz = −∞ 1 σ 2 π exp 1 2 z 2 + μ −∞ 1 2 π exp 1 2 z 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 first term first, we have −∞ 1 σ 2 π 2 exp 1 2 z 2 dx = C −∞ z exp 1 2 z 2 dz = C exp 1 2 z 2 −∞ = 0 .
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