This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Sheen’s 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...
View
Full
Document
This document was uploaded on 12/24/2011.
 Spring '09

Click to edit the document details