kurtosis - 4 ) 4! = µ 4 + 6 µ 2 σ 2 + 3 σ 4 · III....

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MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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± ² ± ² ± ² ± ² ± ² 2.830/6.780J Control of Manufacturing Processes Spring ’08 Handout - Kurtosis MIT OpenCourseWare I. Moment generating function (raw moments): M x ( t ) = e tx f ( x ) dx = E ( e tx ) −∞ t 2 x 2 = (1 + tx + . . . ) f ( x ) dx 2! −∞ II. Normal distribution: 2 2 M x ( t ) = e µt + σ 2 t = (1 + µt + µ 2 t 2 + µ 3 t 3 + µ 4 t 4 . . . ) + (1 + σ 2 t 2 + σ 4 t 4 . . . ) 2! 3! 4! 2 4 2! · µ 0 = 1 µ 1 = µ µ 2 = 2nd raw moment = (coefficient of t 2 ) 2! = µ 2 + σ 2 · µ 3 = 3rd raw moment = (coefficient of t 3 ) 3! = µ 3 + 3 µσ 2 · µ 4 = 4th raw moment = (coefficient of t
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Unformatted text preview: 4 ) 4! = µ 4 + 6 µ 2 σ 2 + 3 σ 4 · III. Kurtosis is deFned as 4th central moment = µ 4 2nd central moment 2 µ 2 2 Using the deFnition n ± ² ³ n µ n = ( − 1) n − j µ j ( µ 1 ) n − j j j =0 , 4 4 4 4 4 µ 4 = ( − 1) 4 · 1 · µ 4 + 1 ( − 1) 3 · µ · µ 3 + 2 ( − 1) 2 · ( µ 2 + σ 2 ) · µ 2 + 3 ( − 1) 1 · ( µ 3 +3 µσ 2 ) · µ 1 + 4 · ( µ 4 +6 µ 2 σ 2 +3 σ 4 ) · 1 Then, for the normal distribution, E ( x − µ ) 4 µ 4 3 σ 4 = = = 3 σ 4 µ 2 2 σ 4 1...
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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kurtosis - 4 ) 4! = µ 4 + 6 µ 2 σ 2 + 3 σ 4 · III....

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