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Slides12-2010

# Slides12-2010 - Normal Random Variables Lecture XII Charles...

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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
Univariate Normal Distribution 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) 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 . 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 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 Substituting into the original integral yields E [ X ] = −∞ 1 σ 2 π ( z σ + 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) Taking the integral of the first term first, we have
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