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# 10_15ans - STAT 410 Examples for Fall 2008 Normal(Gaussian...

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Examples for 10/15/2008 Fall 2008 Normal (Gaussian) Distribution: 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 ) = ( ) - - - dx x x t e e 2 2 ± 2 1 2 π = ( ) - - + dz z z t e e 2 2 2 1 ± = ( ) - - - + dz t z t t e e 2 2 2 2 2 2 1 ± = 2 2 2 ± t t e + , since ( ) 2 2 2 1 t z e - - is the probability density function of a N ( σ t , 1 ) random variable. 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 0 , a 1 , a 2 , … , a n are n + 1 constants, then the random variable U = a 0 + a 1 X 1 + a 2 X 2 + … + a n X n has mean E (U) = a 0 + 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 n are normally distributed random variables, then U is also normally distributed.

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10_15ans - STAT 410 Examples for Fall 2008 Normal(Gaussian...

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