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 ( , 2 ). ( 29 ( 29 , 2 1 exp 2 1 2 2 < < - -- = x x x f Mean and Variance of Normal Distribution Theorem 5.2.1. Let X be distributed N ( , 2 ). Then E [ X ]= and V [ X ]= 2 [ ] ( 29 - -- = dx x x X E 2 2 2 2 1 exp 2 1 dz dx z x x z = + = - = [ ] ( 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 = -- = - = - - - - 2 1 exp 2 1 exp 2 1 exp 2 1 2 2 2 2 z C dz z z C dz z z [ ] ( 29 ( 29 ( 29...
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Lecture 12-2007 - Normal Random Variables Lecture XII...

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