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

12 SAMPLE MOMENTSVar ( Mn2 ) [ (M ) ]=E2 2n − ( EMn) 2 2(50a)= ( n − 1)2 µn 3 4 + (n − 1) (n2 − 2 n +3)σ 4 − n ( n − 1)2 σ 4 (50b)n 3 n 3= µ 4 ( n − 1) 2 + [ ( n − 1)σ ] ( 4 n 2 − 2 n + 3− n ( n − 1) )(50c)= µ 4 ( n − 1) 2 + [ ( n − 1)σ 4 ] ( n 2 − 2 n + 3− n 2 + n )n 3n 3= µ 4 ( n − 1) 2 + [ ( n − 1)σ 4 ] (3 − n )n 3= µ 4 ( n − 1) 2 − [ ( n − 1)σ 4 ] (n − 3)n 3= ( n − 1) 2 µ 4n 3 −( n − 1) (n − 3)σ4n 3(50d)(50e)(50f)5. SAMPLE VARIANCE5.1. Definition of sample variance. The sample variance is defined asS 2 n =1n − 1n∑ (Xi − ¯X) 2n (51)i =1We can write this in terms of moments about the mean asS 2 n =1n − 1n∑i =1(Xi − ¯X n) 2= nn − 1 M 2 n where M 2 n = 1 n(52)n∑ (Xi − ¯X) 2ni=15.2. Expected value of S 2 . We can compute the expected value of S 2 by substituting in from equation 31 asfollowsE ( S 2 n) n =n − 1 E ( )Mn2n n − 1=σ 2(53)n − 1 n= σ 25.3. Variance of S 2 . We can compute the variance of S 2 by substituting in from equation 50 as follows