Lecture 12-2007 - Normal Random Variables Lecture XII...

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Unformatted text preview: Normal Random Variables Lecture XII Univariate Normal Distribution. • Definition 5.2.1. The normal density is given by • When X has the above density, we write symbolically X ~ N ( μ , σ 2 ). ( 29 ( 29 , 2 1 exp 2 1 2 2 ∞ < < ∞- -- = σ σ μ σ π x x x f Mean and Variance of Normal Distribution • Theorem 5.2.1. Let X be distributed N ( μ , σ 2 ). Then E [ X ]= μ and V [ X ]= σ 2 [ ] ( 29 ∫ ∞ ∞- -- = dx x x X E 2 2 2 2 1 exp 2 1 σ μ σ π dz dx z x x z σ μ σ σ μ = + = ⇒- = [ ] ( 29 ∫ ∫ ∫ ∞ ∞- ∞ ∞- ∞ ∞- - + - = - + = dz z dz z z dz z z X E 2 2 2 2 2 1 exp 2 1 2 1 exp 2 1 2 1 exp 2 1 π μ σ σ π μ σ σ π = -- = - = - ∞ ∞- ∞ ∞- ∞ ∞- ∫ ∫ 2 1 exp 2 1 exp 2 1 exp 2 1 2 2 2 2 z C dz z z C dz z z σ σ π [ ] ( 29 ( 29 ( 29 ∫...
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This note was uploaded on 11/08/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 12-2007 - Normal Random Variables Lecture XII...

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