GE 331-Lecture 12 - Normal Distribution A normal r.v. X...

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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 1 Normal Distribution < < = x x x f for 2 ) ( exp 2 1 ) ( 2 2 σ μ π A normal r.v. X with mean μ and variance σ 2 is denoted by X ~ N( μ , σ 2 )
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 2 Normal Distribution (cont) • A normal r.v. with mean μ and variance σ 2 has a symmetric and bell shaped probability density function • It can be verified that • A normal r.v. X with mean μ and variance σ 2 is denoted by X ~ N( μ , σ 2 ) < < = x x x f for 2 ) ( exp 2 1 ) ( 2 2 σ μ π 1 ) ( = +∞ dx x f
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 3 Normal Distribution (cont)
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 4 Standard Normal Distribution •A standard normal r.v. Z has mean 0 and standard deviation 1 : Z ~ N(0, 1) • The pdf for a standard normal distribution is • cdf for a standard normal r.v. is denoted by < < = x x x f for 2 exp 2 1 ) ( 2 π ) ( ) ( ) ( = = Φ z dx x f z Z P z
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 5 Standardization Standardization : for any normal r.v. X ~ N( μ , σ 2 ), • Therefore, ) 1 , 0 ( ~ N X Z σ μ = () ( ) z z Z P x X P x X P Φ = = = ) ( value - z called is where = x z
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 6 ) 1 , 0 ( ~ N X Z σ μ =
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 7 Standardization (cont) Example: X ~ N(3, 4), P(2.6<X<3.6)=? – Standardization: Z=(X-3)/2 ~ N(0,1) •F i n d x such that P(X>x)=1% P(X>x)=P(Z>z)=1% where z=(x-3)/2 z=2.33=(x-3)/2, hence x=7.66 1972 . 0 ) 2 . 0 ( ) 3 . 0 ( ) 2 . 0 ( ) 3 . 0 ( ) 3 . 0 2 . 0 ( ) 2 3 6 . 3 2 3 2 3 6 . 2 ( ) 6 . 3 6 . 2 ( = Φ Φ = < = < < = < < = < < Z P Z P Z P X P X P
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 8 Normal distribution (cont) • What is the probability that X ~ N( μ , σ 2 ) is within one standard deviation around the mean ? μ μ - σ μ + σ 68% 0.68269 ) 1 ( ) 1 ( ) 1 1 ( ) 1 1 ( ) ( = Φ Φ = = = + Z P X P X P
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 9 Normal distribution (cont) • The probability that X is within two standard deviations around the mean • The probability that X is within three standard deviations around the mean % 5 9 ) 2 ( ) 2 ( ) 2 2 ( ) 2 2 ( Φ Φ = = + Z P X P σ μ 99.7% ) 3 ( ) 3 ( ) 3 3 ( ) 3 3 ( Φ Φ = = + Z P X P
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IE 300/GE 331 Lecture 12 Negar Kiyavash, UIUC 10 Normal distribution (cont)
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This note was uploaded on 09/08/2009 for the course GE 331 taught by Professor Negarkayavash during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE 331-Lecture 12 - Normal Distribution A normal r.v. X...

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