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Lecture 12-2008

# Lecture 12-2008 - Normal Random Variables Lecture XII I...

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Normal Random Variables Lecture XII I. Univariate Normal Distribution. A. Definition 5.2.1. The normal density is given by   2 2 1 1 exp , 0 2 2 x f x x         When X has the above density, we write symbolically 2 ~ , X N   . B. Theorem 5.2.1. Let X be 2 , N   . The E X   and 2 V X   . 2 2 2 1 1 exp 2 2 x E X x dx    1. Using the change in variables technique, we create a new random variable z such that x z x z dx dz Substituting into the original integral yields: 2 2 2 2 1 1 exp 2 2 1 1 exp 2 2 1 1 exp 2 2 E X z z dz z z dz z dz      Taking the integral of the first term first, we have: 2 2 2 2 1 1 1 exp exp 2 2 2 1 exp 0 2 z z dz C z z dz C z    

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Lecture 12-2008 - Normal Random Variables Lecture XII I...

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