10 Computational Mechanics of Composite Materials Var(x3=E4E2k2=2o2(o2+2m2) (1.50) =2Var(X)(Var(X)+2E2[X) IⅡnethod Initial algebraic rules can be proved following the method shown below.Using a modified algebraic definition of the variance Var(x2)=Ex-E2x*] (1.51) and the expected value Elx2]=Var(X)+E"[x] (1.52) subtracted from the following equation E2x=(Var(X)+E2[x]=Var2x+2Var(X)E2[x]+E[x] (1.53) we can demonstrate the following desired result: Var(X2)=EX-Var2(X)-2Var(X)E2[X]-E[X] (1.54) III method The characteristic function for the Gaussian PDF has the following form: p()=exp(mit--o2r2）】 (1.55) where p(0=*Ex}5k≥0 (1.56 and =(0)=im (1.57) The mathematical induction rule leads us to the conclusion that pm)=m-to2pm-(0-(0n-1o2.pa-2》0),n≥2 (1.58) which results in the equations10 Computational Mechanics of Composite Materials [ ] [ ] 2 ( )( ( ) 2 [ ]) ( ) 2 ( 2 ) 2 2 4 2 2 2 2 2 Var X Var X E X Var X E X E X m = + = − = σ σ + (1.50) II method Initial algebraic rules can be proved following the method shown below. Using a modified algebraic definition of the variance [] [] 2 4 2 2 Var(X ) = E X − E X (1.51) and the expected value E[X ] Var X E [ ] X 2 2 = ( ) + (1.52) subtracted from the following equation E [ ] X ( ) Var X E [ ] X Var X Var X E [] [] X E X 2 2 4 2 2 2 2 = ( ) + = + 2 ( ) + (1.53) we can demonstrate the following desired result: Var X E[X ] Var X Var X E X E [ ] X 2 4 2 2 4 ( ) = − ( ) − 2 ( ) [ ] − (1.54) III method The characteristic function for the Gaussian PDF has the following form: ( ) 2 2 2 1 ϕ(t) = exp mit − σ t (1.55) where [ ] k k k (0) = i E X ( ) ϕ ; k ≥ 0 (1.56) and ϕ = ϕ (0) ; ϕ′(0) = im (1.57) The mathematical induction rule leads us to the conclusion that ( ) ( ) ( ) ( 1) ( ) ( ) 2 ( 1) 2 ( 2) t im t t n t n n− n− ϕ = − σ ⋅ϕ − − σ ⋅ϕ , 2 n ≥ (1.58) which results in the equations