Slides12-2010 - Normal Random Variables: Lecture XII...

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Normal Random Variables: Lecture XII Charles B. Moss September 20, 2010 Charles B. Moss () Normal Random Variables September 20, 2010 1 / 12
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1 Univariate Normal Distribution Mean Variance 2 Normal and Binomial Distribution Charles B. Moss () Normal Random Variables September 20, 2010 2 / 12
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Univariate Normal Distribution 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 ) 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 . I 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 random variable z such that z = x μ σ x = z σ + μ dx = σ dz (3) Charles B. Moss () Normal Random Variables September 20, 2010 3 / 12
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Continued I Substituting into the original integral yields E [ X ]= Z −∞ 1 σ 2 π ( z σ + 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) Taking the integral of the Frst term Frst, we have Z −∞ 1 σ
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Slides12-2010 - Normal Random Variables: Lecture XII...

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