SAMPLE MOMENTS 1.1. Moments about the origin (raw moments ...

SAMPLE MOMENTS 11⎡( ∑ nE ⎣i =1(Yi − Ȳ ) ) ⎤ ⎡2 (∑ n2⎦ =E ⎣i =1⎡( ∑ n=E ⎣i =1(Y2i − 2 Y i Ȳ + Ȳ 2)) 2 ⎤ ⎦ (47a)Y 2i) ⎤ 2 [⎦ − 2 nE[ 1=nµ 4 + n(n − 1)µ 2 2 − 2nnȲ 2n∑i =1Y 2i]+ n 2 E ( Ȳ 4 ) (47b)[µ4 +(n − 1)µ 2 ] ] [ 12 + n 2 [µ4 +3(n − 1)µ 2 ] ]n 3 2(47c)=nµ 4 + n (n − 1)µ 2 2 − 2 [ µ 4[+ (n − 1) µ 2 ] 1 [2 + µ4n+ 3(n − 1) µ 2 ] ]2(47d)= n2n µ 4 − 2 nn µ 4 + 1 n µ 4 + n2 (n − 1)nµ 2 2−2 n (n − 1)nµ 2 23(n − 1)+ µ 2 2 (47e)n= n2 − 2 n +1µ 4 + (n − 1) (n2 − 2 n +3)µ 2 2nn(47f)= n2 − 2 n +1µ 4 + (n − 1) (n2 − 2 n +3)σ 4nn(47g)Now rewrite equation 41 including 1 n 2as follows⎡( [ ( ) ]E M2 2n = 1 n 2 E ∑ n⎣i =1(Yi − Ȳ ) ) ⎤ 22⎦= 1 ( n 2 − 2 n +1n 2 µ 4 + (n − 1) )(n2 − 2 n +3)σ 4nn(48a)(48b)= n2 − 2 n +1n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 (48c)= ( n − 1)2n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 (48d)Now substitute equations 34 and 48 into equation 33 to obtainWe can simplify this asVar ( Mn2 ) [ (M ) ]=E2 2n − ( EMn2 ) 2(n − 1)2=n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 − ( n − (49)1)2n 2 σ 4