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Unformatted text preview: STAT 410 Examples for 02/22/2008 Spring 2008 Normal (Gaussian) Distribution: mean standard deviation & 2 , & N ( ) ( ) 2 2 & 2 1 2 2 & -- = x e x f , - < x < . ___________________________________________________________________________ EXCEL: ( Z Standard Normal N ( , 1 ) ) = NORMSDIST( z ) gives ( z ) = P( Z z ) = NORMSINV( p ) gives z such that P( Z z ) = p = NORMDIST( x , , , 1 ) gives P( X x ), where X is N ( , 2 ) = NORMDIST( x , , , ) gives f ( x ), p.d.f. of N ( , 2 ) = NORMSINV( p , , ) gives x such that P( X x ) = p , where X is N ( , 2 ) 1. Let X be normally distributed with mean and standard deviation . 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 + b and variance a 2 2 ( standard deviation | a | ). 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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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08