NormalAndExpDist-4

NormalAndExpDist-4 - 1 Normal Random Variable • X is a...

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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 (1777-1855) 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...
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NormalAndExpDist-4 - 1 Normal Random Variable • X is a...

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