10_05ans - STAT 410 Examples for 10/05/2009 Fall 2009...

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Unformatted text preview: STAT 410 Examples for 10/05/2009 Fall 2009 Normal (Gaussian) Distribution . μ – mean σ – standard deviation 2 , σ μ N ( 29 ( 29 2 2 σ μ 2 2 1 σ π-- = x e x f , - ∞ < x < ∞ . Standard Normal Distribution . mean 0 standard deviation 1 N ( , 1 ) Z ~ N ( , 1 ) X ~ N ( μ , σ 2 ) σ μ X Z- = X = μ + σ Z ___________________________________________________________________________ EXCEL: ( Z – Standard Normal N ( , 1 ) ) = NORMSDIST( z ) gives Φ ( z ) = P( Z ≤ z ) = NORMSINV( p ) gives z such that P( Z ≤ z ) = p = NORMDIST( x , μ , σ , 1 ) gives P( X ≤ x ), where X is N ( μ , σ 2 ) = NORMDIST( x , μ , σ , ) gives f ( x ), p.d.f. of N ( μ , σ 2 ) = NORMSINV( p , μ , σ ) gives x such that P( X ≤ x ) = p , where X is N ( μ , σ 2 ) ___________________________________________________________________________ 1. Let X be normally distributed with mean μ and standard deviation σ . Find the moment-generating function of X, M X ( t ). M X ( t ) = E ( e t X ) = ( 29 ∫ ∞ ∞--- ⋅ dx x x t e e 2 2 σ μ 2 1 2 σ π = ( 29 ∫ ∞ ∞-- + ⋅ dz z...
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This note was uploaded on 04/15/2010 for the course STAT 410 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_05ans - STAT 410 Examples for 10/05/2009 Fall 2009...

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