> restart:
> Digits:=20:
> with(Statistics):
> G:=RandomVariable(Normal(0,1)):
> X1:=RandomVariable(Uniform(-1,2)):
> CDF1:=CDF(X1,x1);
> X2:=RandomVariable(Gamma(1/2,2)):
> CDF2:=CDF(X2,x2);
> X3:=RandomVariable(Beta(r,t-r)):
> CDF3:=subs(t3=((x3+1)/3),r=2,t=5,CDF(X3,t3));

                        {     0              x1 < -1
                        {
                        {        x1
                CDF1 := { 1/3 + ----         x1 < 2
                        {        3
                        {
                        {     1             otherwise


              {                 0                        x2 <= 0
      CDF2 := {
              { -exp(-2 x2) - 2 exp(-2 x2) x2 + 1        0 < x2


          {
          {
  CDF3 := {      x3
          { 0 , ---- < -1/3
          {      3

            / x3       \2                          x3
            |---- + 1/3|  hypergeom([2, -2], [3], ---- + 1/3)
            \ 3        /                           3
        1/2 ------------------------------------------------- ,
                               Beta(2, 3)

         x3
        ---- < 2/3
         3

        1 , otherwise

> X1:=evalf(subs(subs(x1=1,CDF1)));
> X2:=evalf(subs(subs(x2=1,CDF2)));
> X3:=evalf(subs(subs(x3=1,CDF3)));

                     X1 := 0.66666666666666666667


                     X2 := 0.59399415029016192433


                     X3 := 0.88888888888888888889

> u1:=fsolve(CDF(G,x)=X1,x);
> u2:=fsolve(CDF(G,x)=X2,x);
> u3:=fsolve(CDF(G,x)=X3,x);

                     u1 := 0.43072729929545749022


                     u2 := 0.23783161172327289985


                     u3 := 1.2206403488473496983

> INEC_g1:=evalf(subs(x=u1,PDF(G,x)));
> INEC_g2:=evalf(subs(x=u2,PDF(G,x)));
> INEC_g3:=evalf(subs(x=u3,PDF(G,x)));

                  INEC_g1 := 0.36359977467531772230


                  INEC_g2 := 0.38781747667892138429


                  INEC_g3 := 0.18939510112030681087

> INEC_h1:=evalf(subs(x=u1,diff(PDF(G,x),x)));
> INEC_h2:=evalf(subs(x=u2,diff(PDF(G,x),x)));
> INEC_h3:=evalf(subs(x=u3,diff(PDF(G,x),x)));

                  INEC_h1 := -0.15661234897033648136


                 INEC_h2 := -0.092235255533000673534


                  INEC_h3 := -0.23118330230147037728

> IMT_g1:=1/evalf(subs(x1=1,diff(CDF1,x1)));
> IMT_g2:=1/evalf(subs(x2=1,diff(CDF2,x2)));
> IMT_g3:=1/evalf(subs(x3=1,diff(CDF3,x3)));

                   IMT_g1 := 3.0000000000000000000


                   IMT_g2 := 1.8472640247326625569


                   IMT_g3 := 3.3750000000000000001

> IMT_h1:=-evalf(subs(x1=1,diff(CDF1,x1$2)))/evalf(subs(x1=1,diff(CDF1,x
> 1)))^3;
> IMT_h2:=-evalf(subs(x2=1,diff(CDF2,x2$2)))/evalf(subs(x2=1,diff(CDF2,x
> 2)))^3;
> IMT_h3:=-evalf(subs(x3=1,diff(CDF3,x3$2)))/evalf(subs(x3=1,diff(CDF3,x
> 3)))^3;

                            IMT_h1 := -0.


                   IMT_h2 := 3.4123843770715149426


                   IMT_h3 := 17.085937500000000002

> diff(f(g(x)),x$2);

                                                  / 2      \
            (2)           /d      \2              |d       |
          (D   )(f)(g(x)) |-- g(x)|  + D(f)(g(x)) |--- g(x)|
                          \dx     /               |  2     |
                                                  \dx      /

> IMT_h1*INEC_g1^2+IMT_g1*INEC_h1;
> IMT_h2*INEC_g2^2+IMT_g2*INEC_h2;
> IMT_h3*INEC_g3^2+IMT_g3*INEC_h3;

                       -0.46983704691100944408


                        0.34284791435655692599


                       -0.16736245021943200214

> 
