Mathematical Preliminaries 9 1,=jre号d:l,=jre9d;l,=jre号d: (1.41) I=jted:Is=jed It should be mentioned that the values of the odd integrals on the real domain are equal to 0 in the following calculation jf(x)g(x)dx-jf()g(dx+jf(x)g(x)d (1.42) If the function fx)is odd and g(x)is even x)=-fx),g(x)=g(x), (1.43) then it can be written jrd--d--d (1.44) Considering that the odd indices integrals are calculated;this results in 1,=je片di=际 (1.45) 5=rah=-了eh=-de5, (1.46) ∫e d=V2 4a-了-. (1.47) =-a-[-w际 After simplification the result is Ex4]=30+602m2+m=E+[X ]+6Var(X)E2[X]+3Var2(X) (1.48) Ex2=o2+m2=E2[X]+Var(X) (1.49)Mathematical Preliminaries 9 I t e dt t ∫ +∞ −∞ − = 2 2 4 1 ; I t e dt t ∫ +∞ −∞ − = 2 2 3 2 ; I t e dt t ∫ +∞ −∞ − = 2 2 2 3 ; I te dt t ∫ +∞ −∞ − = 2 2 4 ; I e dt t ∫ +∞ −∞ − = 2 2 5 (1.41) It should be mentioned that the values of the odd integrals on the real domain are equal to 0 in the following calculation ∫ ∫ ∫ +∞ −∞ +∞ −∞ = + 0 0 f (x)g(x)dx f (x)g(x)dx f (x)g(x)dx (1.42) If the function f(x) is odd and g(x) is even f(-x)=-f(x), g(-x)=g(x), (1.43) then it can be written ∫ ∫ ∫ +∞ +∞ −∞ = − = − 0 0 0 f (x)g(x)dx f ( x)g(x)dx f (x)g(x)dx . (1.44) Considering that the odd indices integrals are calculated; this results in 2π 2 2 5 = ∫ = +∞ −∞ − I e dt t (1.45) 2π ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 3 = − + = = = − = − ∫ ∫ ∫ ∫ +∞ −∞ − +∞ −∞ − − +∞ −∞ +∞ −∞ − +∞ −∞ − te e dt I t e dt t te dt td e t t t t t (1.46) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ∫ = − ∫ = − − ∫ +∞ −∞ − +∞ −∞ − − +∞ −∞ +∞ −∞ 4 − 3 3 3 1 2 2 2 2 2 2 2 2 I t e dt t de t e e dt t t t t 3 3 3 3 2π 2 2 2 2 2 2 2 2 2 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ∫ = − ∫ = − − ∫ +∞ −∞ − +∞ −∞ − − +∞ −∞ +∞ −∞ − t e dt tde te e dt t t t t . (1.47) After simplification the result is [ ] 3 6 [] [] 6 ( ) 3 ( ) 4 4 2 2 4 4 2 2 E X = σ + σ m + m = E X + Var X E X + Var X (1.48) [ ] [ ] ( ) 2 2 2 2 E X = σ + m = E X +Var X (1.49)