> restart:
> with(Statistics):
> assume(u<1,u>0,r>0,t>r);
> beta_:=RandomVariable(Beta(r,t-r)):
> pdf:=factor(subs(u=(x-a)/(b-a),PDF(beta_,u))/(b-a));
> ddf:=(diff(pdf,x));
> cf:=simplify(subs(w=(u-a)/(b-a),CharacteristicFunction(beta_,w)),hyper
> geom);
> factor(ddf/pdf);
> cdf:=subs(u=(x-a)/(b-a),CDF(beta_,u));
> mu_:=a+(b-a)*Mean(beta_);
> var_:=(b-a)^2*Variance(beta_);
> skew_:=simplify(convert(Skewness(beta_),GAMMA),symbolic);
> kurt_:=simplify(convert(Kurtosis(beta_),GAMMA),symbolic);
> sol:=subs(m='mu',v='var',solve({mu_=m,var_=v},{r,t}));
> qdf:=a+(b-a)*Quantile(beta_,p);
> qdf2:=solve(cdf=p,x);
> pdfgr:=map(_x->factor(subs(t=r+s,_x)),[diff(pdf, r)/pdf,diff(pdf,
> t)/pdf,diff(pdf, a)/pdf,diff(pdf, b)/pdf]);
> cdfgr:=[diff(cdf, r),diff(cdf, t),diff(cdf, a),diff(cdf, b)];
> valnum:=r=2,t=5,a=-1,b=2:
> evalf(subs(valnum,x=1,ddf));
> evalf(subs(valnum,x=1,pdf));
> evalf(subs(valnum,x=1,cdf));
> evalf(subs(valnum,x=1,map(_x->_x*pdf,pdfgr)));
> evalf(subs(valnum,x=1,cdfgr));
> evalf(solve(subs(valnum,cdf)=0.95,x));
> evalf(subs(valnum,mu_));
> evalf(subs(valnum,sqrt(var_)));
> evalf(subs(valnum,skew_));
> evalf(subs(valnum,kurt_));
> evalf(subs(valnum,var_));
> evalf(subs(valnum,[mu_,sqrt(var_)]));
> evalf(subs(sol,mu=mu_,var=var_,valnum,[r,t]));
> 

                    /-x + a\(-1 + r~) /-b + x\(t~ - r~ - 1)
                    |------|          |------|
                    \-b + a/          \-b + a/
           pdf := - ---------------------------------------
                          Beta(r~, t~ - r~) (-b + a)


         /-x + a\(-1 + r~)           /-b + x\(t~ - r~ - 1)
         |------|          (-1 + r~) |------|
         \-b + a/                    \-b + a/
  ddf := -------------------------------------------------
                (-x + a) Beta(r~, t~ - r~) (-b + a)

           /-x + a\(-1 + r~) /-b + x\(t~ - r~ - 1)
           |------|          |------|              (t~ - r~ - 1)
           \-b + a/          \-b + a/
         - -----------------------------------------------------
                    (-b + x) Beta(r~, t~ - r~) (-b + a)


                                           (u~ - a) I
               cf := hypergeom([r~], [t~], ----------)
                                             b - a


                b - 2 x - r~ b + a + t~ x - t~ a + r~ a
              - ---------------------------------------
                           (-b + x) (-x + a)


           /x - a\r~                                        x - a
           |-----|   hypergeom([r~, 1 - t~ + r~], [1 + r~], -----)
           \b - a/                                          b - a
    cdf := -------------------------------------------------------
                            Beta(r~, t~ - r~) r~


                                   (b - a) r~
                        mu_ := a + ----------
                                       t~


                                   2
                            (b - a)  r~ (t~ - r~)
                    var_ := ---------------------
                                  2
                                t~  (t~ + 1)


                                      1/2
                            2 (t~ + 1)    (2 r~ - t~)
                skew_ := - ---------------------------
                             1/2          1/2
                           r~    (t~ - r~)    (2 + t~)


                 2          2        2                 2
            3 (r~  t~ - 6 r~  - r~ t~  + 6 r~ t~ - 2 t~ ) (t~ + 1)
   kurt_ := ------------------------------------------------------
                       r~ (-t~ + r~) (3 + t~) (2 + t~)


                                      2
                 b a - a mu + var + mu  - mu b             2    2
  sol := {t~ = - -----------------------------, r~ = - (b a  - a  mu
                              var

                         2                  2              3
         + a var + 2 a mu  - 2 b a mu + b mu  - var mu - mu )/(

        (-b + a) var)}


                             qdf := FAIL


  qdf2 := RootOf(-

        /  _Z - a\r~                                          _Z - a
        |- ------|   hypergeom([r~, 1 - t~ + r~], [1 + r~], - ------)
        \  -b + a/                                            -b + a

         + p Beta(r~, t~ - r~) r~)


               -x + a       -b + x
  pdfgr := [ln(------) - ln(------) - Psi(r~) + Psi(s),
               -b + a       -b + a

           -b + x
        ln(------) - Psi(s) + Psi(r~ + s),
           -b + a

          -b + x + r~ b - r~ x - s x + s a
        - --------------------------------,
                 (-x + a) (-b + a)

          x + r~ b - r~ x - s x + s a - a
        - -------------------------------]
                 (-b + x) (-b + a)


              /x - a\r~
              |-----|   %1 (Psi(r~) - Psi(t~ - r~))
              \b - a/
  cdfgr := [- -------------------------------------
                      Beta(r~, t~ - r~) r~

               /x - a\r~           /x - a\r~    x - a
               |-----|   %1        |-----|   ln(-----) %1
               \b - a/             \b - a/      b - a
         - --------------------- + ----------------------
                               2    Beta(r~, t~ - r~) r~
           Beta(r~, t~ - r~) r~

            /x - a\r~ / d    \
            |-----|   |--- %1|
            \b - a/   \dr~   /
         + --------------------,
           Beta(r~, t~ - r~) r~

          /x - a\r~
          |-----|   %1 (Psi(t~ - r~) - Psi(t~))
          \b - a/
        - -------------------------------------
                  Beta(r~, t~ - r~) r~

            /x - a\r~ / d    \
            |-----|   |--- %1|
            \b - a/   \dt~   /
         + --------------------,
           Beta(r~, t~ - r~) r~

        /x - a\r~ /    1      x - a  \
        |-----|   |- ----- + --------| (b - a) %1
        \b - a/   |  b - a          2|
                  \          (b - a) /              /x - a\r~
        ----------------------------------------- + |-----|
                Beta(r~, t~ - r~) (x - a)           \b - a/

        (1 - t~ + r~)

                                                   x - a
        hypergeom([1 + r~, 2 - t~ + r~], [2 + r~], -----)
                                                   b - a

        /    1      x - a  \
        |- ----- + --------|/(Beta(r~, t~ - r~) (1 + r~)),
        |  b - a          2|
        \          (b - a) /

                /x - a\r~
                |-----|   %1
                \b - a/               /x - a\r~
        - ------------------------- - |-----|   (1 - t~ + r~)
          Beta(r~, t~ - r~) (b - a)   \b - a/

                                                   x - a
        hypergeom([1 + r~, 2 - t~ + r~], [2 + r~], -----) (x - a)
                                                   b - a

           /                                    2
          /  (Beta(r~, t~ - r~) (1 + r~) (b - a) )]
         /

                                               x - a
  %1 := hypergeom([r~, 1 - t~ + r~], [1 + r~], -----)
                                               b - a


                            -0.4444444445


                             0.2962962962


                             0.8888888888


  [0.08010750986 + 0.2962962962 Psi(s), -0.3255147523

         - 0.2962962962 Psi(s) + 0.2962962962 Psi(2. + s),

        -0.04938271604 + 0.09876543209 s,

        -0.3950617284 + 0.1975308642 s]


  [-0.3604134293 + 2.666666666 diff(hypergeom([2, -2], [3], 2/3), 2),

        0.5185185182

         + 2.666666666 diff(hypergeom([2, -2], [3], 2/3), 5),

        -0.0987654321, -0.1975308640]


                             1.254186123


                             0.2000000000


                             0.6000000000


                             0.2857142858


                             2.357142857


                             0.3600000000


                     [0.2000000000, 0.6000000000]


                               [2., 5.]

> factor(subs(sol,r/t));

                                a - mu
                                ------
                                -b + a

> R:=map(simplify,collect(-(b-2*x-r*b+a+t*x-t*a+r*a)/(-b+x)/(-x+a),t));

                      t~     b - 2 x - r~ b + a + r~ a
               R := ------ - -------------------------
                    -b + x       (-b + x) (-x + a)

> factor(R-(-(t-r-1)/(b-x)+(r-1)/(x-a)));

                                  0

> kurt_;

             2          2        2                 2
        3 (r~  t~ - 6 r~  - r~ t~  + 6 r~ t~ - 2 t~ ) (t~ + 1)
        ------------------------------------------------------
                   r~ (-t~ + r~) (3 + t~) (2 + t~)

> A:=collect(-(r^2*t-6*r^2-r*t^2+6*r*t-2*t^2),r);

                             2      2                  2
             A := (6 - t~) r~  + (t~  - 6 t~) r~ + 2 t~

> ((-2-r)*t+r*(r+6)/2)^2/(-2-r)-r^2*(r-6)^2/(4*(2+r))-12*r^2/(2+r);

        /               r~ (r~ + 6)\2
        |(-2 - r~) t~ + -----------|      2         2        2
        \                    2     /    r~  (r~ - 6)    12 r~
        ----------------------------- - ------------- - ------
                   -2 - r~                8 + 4 r~      2 + r~

> factor((6-t)*r^2+(t^2-6*t)*r);

                       -r~ (-6 + t~) (-t~ + r~)

> S:=simplify(int(pdf^2,x=a..b),symbolic);

               (2 - 2 t~)
       (-b + a)
  S := ------------------
                        2
       Beta(r~, t~ - r~)

           b
          /
         |           (-2 + 2 r~)         (2 t~ - 2 r~ - 2)
         |   (-x + a)            (-b + x)                  dx
         |
        /
          a

> delta:=S-Beta(2*r-1,2*t-2*r-1)/Beta(r,t-r)^2/(b-a);

                   (2 - 2 t~)
           (-b + a)
  delta := ------------------
                            2
           Beta(r~, t~ - r~)

           b
          /
         |           (-2 + 2 r~)         (2 t~ - 2 r~ - 2)
         |   (-x + a)            (-b + x)                  dx
         |
        /
          a

           Beta(-1 + 2 r~, 2 t~ - 2 r~ - 1)
         - --------------------------------
                               2
              Beta(r~, t~ - r~)  (b - a)

> evalf(subs(r=2.5,t=4.9,a=-1.2,b=3.5,delta));

                         -9                  -10
                   0.1 10   - 0.2353302428 10    I

> int(exp(I*u*x)*pdf,x=-infinity..infinity);

     infinity               /-x + a\(-1 + r~) /-b + x\(t~ - r~ - 1)
    /           exp(u~ x I) |------|          |------|
   |                        \-b + a/          \-b + a/
   |          - --------------------------------------------------- d
   |                        Beta(r~, t~ - r~) (-b + a)
  /
    -infinity

        x

> 
