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

SAMPLE MOMENTS 15( )E(X − µ ) 4 = d4dt 4 e t2 σ 22| t =0(t 3 σ 6 ( e t2 σ 22) ( ))+3tσ 4 e t2 σ 22| t =0= d dt( ( ) ( ) ( ) ( ))= t 4 σ 8 e t2 σ 22 +3t 2 σ 6 e t2 σ 22 +3t 2 σ 6 e t2 σ 22 +3σ 4 e t2 σ 2(63)2 | t =0( ( ) ( ) ( ))= t 4 σ 8 e t2 σ 22 +6t 2 σ 6 e t2 σ 22 +3σ 4 e t2 σ 22 | t =0=3σ 46.2. Variance of S 2 . Let X 1 ,X 2 ,...X n be a random sample from a normal population with mean µ andvariance σ 2 < ∞.We know from equation 54 thatVar ( Sn2 ) n 2=( n − 1) Var( M 2 )2 nn 2( ( n − 1) 2 µ 4=( n − 1) 2 n 3−)( n − 1) (n − 3) σ4n 3(64)= µ 4n−(n − 3)σ4n ( n − 1)If we substitute in for µ 4 from equation 63 we obtainVar ( S 2 n) =µ 4n−(n − 3)σ4n (n − 1)= 3 σ4n−(n − 3)σ4n ( n − 1)(3(n − 1)− (n − 3))σ4=n ( n − 1)(3n − 3 − n +3))σ4=n ( n − 1)(65)= 2 nσ4n ( n − 1)6.3. Variance of ˆσ 2 . It is easy to show that= 2 σ4( n − 1)Var(ˆσ 2 )= 2 σ4 ( n − 1)n 2 (66)