elemprob-page16

elemprob-page16 - A synonym for normal is Gaussian The rst...

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A synonym for normal is Gaussian. The first thing to do is show that this is a density. Let I = 0 e - x 2 / 2 dx. Then I 2 = 0 -∞ e - x 2 / 2 e - y 2 / 2 dx dy. Changing to polar coordinates, I 2 = π/ 2 0 0 re - r 2 / 2 dr = π/ 2 . So I = π/ 2, hence -∞ e - x 2 / 2 dx = 2 π as it should. Note xe - x 2 / 2 dx = 0 by symmetry, so E Z = 0. For the variance of Z , we use integration by parts: E Z 2 = 1 2 π x 2 e - x 2 / 2 dx = 1 2 π x · xe - x 2 / 2 dx. The integral is equal to - xe - x 2 / 2 -∞ + e - x 2 / 2 dx = 2 π. Therefore Var Z = E Z 2 = 1. We say X is a N ( μ, σ 2 ) if X = σZ + μ , where Z is a N (0 , 1). We see that F X ( x ) = P ( X x ) = P ( μ + σZ x ) = P ( Z ( x - μ ) ) = F Z (( x - μ ) ) if σ > 0. (A similar calculation holds if σ < 0.) Then by the chain rule X has density f X ( x ) = F X ( x ) = F Z (( x - μ ) ) = 1 σ f Z (( x - μ ) ) . This is equal to 1 2 πσ e - ( x - μ ) 2 / 2 σ 2 . E X = μ + E Z and Var X = σ 2 Var Z , so E X = μ, Var X = σ 2 . If X is N ( μ, σ 2 ) and Y = aX + b , then Y = a ( μ + σZ )+
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