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Unformatted text preview: 1 Normal Random Variable X is a Normal Random Variable : X ~ N( m , s 2 ) Probability Density Function (PDF): Also called Gaussian Note: f(x) is symmetric about m Common for natural phenomena: heights, weights, etc. Often results from the sum of multiple variables x e x f x where 2 1 ) ( 2 2 2 / ) ( s m s m ] [ X E 2 ) ( s X Var ) ( x f x m Carl Friedrich Gauss Carl Friedrich Gauss (17771855) was a remarkably influential German mathematician Started doing groundbreaking math as teenager Did not invent Normal distribution, but popularized it He looked more like Martin Sheen Who is, of course, Charlie Sheens father Properties of Normal Random Variable Let X ~ N( m , s 2 ) Let Y = a X + b Y ~ N( a m + b , a 2 s 2 ) E[Y] = E[ a X + b ] = a E[X] + b = a m + b Var(Y) = Var( a X + b ) = a 2 Var(X) = a 2 s 2 Differentiating F Y ( x ) w.r.t. x , yields f Y ( x ) , the PDF for y : Special case: Z = (X m )/ s ( a = 1/ s , b = m / s ) Z ~ N( a m + b , a 2 s 2 ) = N( m / s m / s , (1/ s ) 2 s 2 ) = N(0, 1) ) ( ) ( ) ( ) ( ) ( a b x a b x X Y F X P x b aX P x Y P x F ) ( ) ( ) ( ) ( 1 a b x a a b x dx d dx d X X Y Y f F x F x f Standard (Unit) Normal Random Variable Z is a Standard (or Unit) Normal RV : Z ~ N(0, 1) E[Z] = m = 0 Var(Z) = s 2 = 1 SD(Z) = s = 1 CDF of Z, F Z ( z ) does not have closed form We denote F Z ( z ) as ( z ): phi of z By symmetry: ( z ) = P( Z z ) = P( Z z ) = 1 ( z ) Use Z to compute X ~ N( m , s 2 ), where s > 0 Table of ( z ) values in textbook, p. 201 and handout dx e dx e z Z P z z x z x 2 / 2 / ) ( 2 2...
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 Spring '09

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