02_22 - STAT 410 Examples for Spring 2008 Normal(Gaussian...

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Unformatted text preview: STAT 410 Examples for 02/22/2008 Spring 2008 Normal (Gaussian) Distribution: μ – mean σ – standard deviation & ¡ ¢ £ ¤ ¥ 2 , & ¡ N ( ) ( ) 2 2 & ¡ 2 1 2 2 & ¢-- = x e x f , - ∞ < x < ∞ . ___________________________________________________________________________ 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 ). Let Y = a X + b . Then M Y ( t ) = e b t M X ( a t ). Therefore, Y is normally distributed with mean a μ + b and variance a 2 σ 2 ( standard deviation | a | σ ). If X 1 , X 2 , … , X n are n independent random variables and a , a 1 , a 2 , … , a n are n + 1 constants, then the random variable U = a + a 1 X 1 + a 2 X 2 + … + a n X n has mean E (U) = a + a 1 E (X 1 ) + a 2 E (X 2 ) + … + a n E (X n ) and variance Var (U) = a 1 2 Var (X 1 ) + a 2 2 Var (X 2 ) + … + a n 2 Var (X n ) If X 1 , X 2 , … , X n are normally distributed random variables, then U is also normally...
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02_22 - STAT 410 Examples for Spring 2008 Normal(Gaussian...

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