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

14 SAMPLE MOMENTS6. NORMAL POPULATIONS6.1. Central moments of the normal distribution. For a normal population we can obtain the central momentsby differentiating the moment generating function. The moment generating function for the centralmoments is as followsM X (t) =e t2 σ 22 . (59)The moments are then as follows. The first central moment isE(X − µ) = d dt) (e t2 σ 22( )= tσ 2 e t2 σ 22| t =0| t =0(60)=0The second central moment is( )E(X − µ ) 2 = d2e t2 σ 2dt 2 2 | t =0= d (t ( ))σ 2 e t2 σ 22 | t =0dt( ( ) ( (61)))= t 2 σ 4 e t2 σ 22 + σ 2 e t2 σ 22 | t =0= σ 2The third central moment is( )E(X − µ ) 3 = d3dt 3 e t2 σ 22= d dt==| t =0(t 2 σ 4 ( e t2 σ 22(t 3 σ 6 ( e t2 σ 22(t 3 σ 6 ( e t2 σ 22) ( ))+ σ 2 e t2 σ 22)+2tσ 4 ( e t2 σ 22) ( ))+3tσ 4 e t2 σ 22| t =0) ( ))+ tσ 4 e t2 σ 22| t =0| t =0(62)=0The fourth central moment is