8 Computational Mechanics of Composite Materials probability density functions of the input frequently takes place in which numerous engineering problems. I method Starting from the definition of the variance of a ny random variable one can write Var(Y)=E(Y2)-E2(Y) (1.34) Let Y=X2,then Var(X2)=E(X2)2)-E2(X2) (1.35) The value of E will be determined through integration of the characteristic function for the Gaussian probability density function (1.36) where m=E[X]and o=Var(X)denote the expected value and standard deviation of the considered distribution,respectively.Next,the following standardised variable is introduced t=x-m,where x=to+m,dx=adt (1.37) which gives k了+mreu叫} (1.38) After some algebraic transforms of the integrand function it is obtained that Ek☆了(or+4onmn户+6anm+4omr+me号h (1.39) and,dividing into particular integrals,there holds Ek]应oh+4oml,+6o2m21,+4om14+m1,)e号 (1.40) where the components denote8 Computational Mechanics of Composite Materials probability density functions of the input frequently takes place in which numerous engineering problems. I method Starting from the definition of the variance of a ny random variable one can write ( ) ( ) ( ) 2 2 Var Y = E Y − E Y (1.34) Let 2 Y = X , then ( ) (( ) ) ( ) 2 2 2 2 2 Var X = E X − E X (1.35) The value of [ ] 4 E X will be determined through integration of the characteristic function for the Gaussian probability density function [ ] ∫ +∞ −∞ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − dx x m E X x 2 2 4 2 4 1 2 ( ) exp σ π σ (1.36) where m=E[X] and σ = Var(X ) denote the expected value and standard deviation of the considered distribution, respectively. Next, the following standardised variable is introduced σ x m t − = , where x = tσ + m,dx = σdt (1.37) which gives [ ] dt t E X ∫ t m +∞ −∞ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + − 2 ( ) exp 2 4 2 4 1 σ π (1.38) After some algebraic transforms of the integrand function it is obtained that E[ ] X t mt m t m t m e dt t ∫ +∞ −∞ − = + + + + 2 2 ( 4 6 4 ) 4 4 3 3 2 2 2 3 4 2 4 1 σ σ σ σ π (1.39) and, dividing into particular integrals, there holds [ ] 2 2 ( 4 6 4 ) 5 4 4 3 3 2 2 2 3 1 4 2 4 1 t E X I mI m I m I m I e − = σ + σ + σ + σ + π (1.40) where the components denote