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

8 SAMPLE MOMENTS[ (M )2 2]E n⎡(=E ⎣⎡(=E ⎣⎡(=E ⎣1n1n1n= 1 n 2 E ⎡⎣) ⎤ 2 n∑ (Xi − ¯X) 2n⎦i =1n∑i =1(Xi − ¯X ) ) ⎤ 22⎦n∑ (Yi − Ȳ ) ) ⎤ 22⎦i =1( n∑i =1(Yi − Ȳ ) ) ⎤ 22⎦(41a)(41b)(41c)(41d)Ignoring 1 n 2for now, expand equation 41 as follows⎡( ∑ nE ⎣i =1(Yi − Ȳ ) ) ⎤ 22⎡( ∑ n⎦ =E ⎣i =1⎡( ∑ n= E ⎣i =1⎡({ ∑ n= E ⎣i =1⎡({ ∑ n= E ⎣i =1⎡( ∑ n= E ⎣i =1⎡( ∑ n= E ⎣i =1(Y2i − 2 Y i Ȳ + Ȳ 2)) 2 ⎤ ⎦ (42a)Yi2 − 2 ȲY 2iY 2iY 2iY 2in∑i =1Y i +n∑i =1} ) ⎤ 2− 2 n Ȳ 2 + n Ȳ 2 ⎦}) ⎤ 2− nȲ 2 ⎦) 2− 2 n Ȳ 2 n ∑) ⎤ 2 [⎦ − 2 nEi =1Ȳ 2Y 2in∑i =1Ȳ 2 ) 2⎤⎦+ n 2 Ȳ 4 ⎤⎦Y 2i(42b)(42c)(42d)(42e)]+ n 2 E ( Ȳ 4) (42f)Now consider the first term on the right of 42 which we can write as