> restart:
> with(Statistics):
> assume(beta>0,beta,real,alpha,real):
> logistic_:=RandomVariable(Logistic(alpha,beta)):
> pdf:=PDF(logistic_,x);
> ddf:=factor(diff(pdf,x));
> cdf:=CDF(logistic_,x);
> cf:=subs(u=x,CharacteristicFunction(logistic_,u));
> rough:=factor(int(pdf^2,x=-infinity..infinity));
> mu_:=Mean(logistic_);
> var_:=Variance(logistic_);
> #skew_:=simplify(convert(Skewness(logistic_),GAMMA),symbolic);
> skew_:=Int((x-mu_)^3*pdf,x=-infinity..infinity)/var_^(3/2);
> #kurt_:=simplify(convert(Kurtosis(logistic_),GAMMA),symbolic);
> kurt_:=Int((x-mu_)^4*pdf,x=-infinity..infinity)/var_^2;
> qdf:=Quantile(logistic_,p);
> qdf2:=solve(cdf=p,x);
> pdfgr:=map(factor,[diff(pdf,alpha),diff(pdf,beta)]);
> factor(pdfgr[2]-(x-alpha)*pdfgr[1]/beta);
> cdfgr:=map(factor,[diff(cdf,alpha),diff(cdf,beta)]);
> factor(dcdf[2]/dcdf[1]);
> valnum:=alpha=-1/2,beta=3/2:
> evalf(subs(valnum,x=1,ddf));
> evalf(subs(valnum,x=1,pdf));
> evalf(subs(valnum,x=1,cdf));
> evalf(subs(valnum,x=1,cf));
> evalf(subs(valnum,x=1,map(_x->_x,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_)]));

                                  x - alpha~
                              exp(----------)
                                    beta~
                 pdf := ----------------------------
                              /        x - alpha~ \2
                        beta~ |1 + exp(----------)|
                              \          beta~    /


                        x - alpha~  /         x - alpha~ \
                    exp(----------) |-1 + exp(----------)|
                          beta~     \           beta~    /
           ddf := - --------------------------------------
                             2 /        x - alpha~ \3
                        beta~  |1 + exp(----------)|
                               \          beta~    /


                                      1
                     cdf := ---------------------
                                      x - alpha~
                            1 + exp(- ----------)
                                        beta~


                         exp(alpha~ x I) Pi beta~ x
                   cf := --------------------------
                              sinh(Pi beta~ x)


                                       1
                           rough := -------
                                    6 beta~


                            mu_ := alpha~


                                       2   2
                                  beta~  Pi
                          var_ := ----------
                                      3


  skew_ :=

                           infinity
                          /                     3     x - alpha~
               1/2       |          (x - alpha~)  exp(----------)
            3 3          |                              beta~
        ---------------  |          ----------------------------- dx
              2   2 3/2  |                /        x - alpha~ \2
        (beta~  Pi )     |          beta~ |1 + exp(----------)|
                        /                 \          beta~    /
                          -infinity


                          infinity
                         /                     4     x - alpha~
                        |          (x - alpha~)  exp(----------)
                9       |                              beta~
   kurt_ := ----------  |          ----------------------------- dx
                 4   4  |                /        x - alpha~ \2
            beta~  Pi   |          beta~ |1 + exp(----------)|
                       /                 \          beta~    /
                         -infinity


                                              p
                   qdf := alpha~ + beta~ ln(-----)
                                            1 - p


                                       -1 + p
                 qdf2 := alpha~ - ln(- ------) beta~
                                         p


                x - alpha~  /         x - alpha~ \
            exp(----------) |-1 + exp(----------)|
                  beta~     \           beta~    /      x - alpha~  /
  pdfgr := [--------------------------------------, exp(----------) |
                     2 /        x - alpha~ \3             beta~     \
                beta~  |1 + exp(----------)|
                       \          beta~    /

                   x - alpha~                        x - alpha~
        -x + x exp(----------) + alpha~ - alpha~ exp(----------)
                     beta~                             beta~

                             x - alpha~ \   / /     3
         - beta~ - beta~ exp(----------)|  /  |beta~
                               beta~    / /   \

        /        x - alpha~ \3\
        |1 + exp(----------)| |]
        \          beta~    / /


                                x - alpha~
                            exp(----------)
                                  beta~
                   - -----------------------------
                          2 /        x - alpha~ \2
                     beta~  |1 + exp(----------)|
                            \          beta~    /


                          x - alpha~
                    exp(- ----------)
                            beta~
  cdfgr := [- ------------------------------,
              /          x - alpha~ \2
              |1 + exp(- ----------)|  beta~
              \            beta~    /

                             x - alpha~
          (x - alpha~) exp(- ----------)
                               beta~
        - -------------------------------]
          /          x - alpha~ \2      2
          |1 + exp(- ----------)|  beta~
          \            beta~    /


                               dcdf[2]
                               -------
                               dcdf[1]


                            -0.04038122117


                             0.1310746222


                             0.7310585787


                   0.07430698329 - 0.04059408999 I


                   [0.04038122117, -0.04700186026]


                    [-0.1310746222, -0.1310746222]


                             3.916658469


                            -0.5000000000


                             2.720699048


                                         -11
                          0.1330935130 10


                             4.199999999


                             7.402203303


                     [-0.5000000000, 2.720699048]

> qdf2 := alpha-ln(-(-1+p)/p)*beta;
> diff(kurt_,alpha);
> diff(kurt_,beta);

                                       -1 + p
                 qdf2 := alpha~ - ln(- ------) beta~
                                         p


                                  0


                infinity
               /                        5                     4
      9       |          (-_t0 + alpha~)  %1   (-_t0 + alpha~)  %1
  ----------  |          ------------------- - -------------------
       4   4  |                3         2           2         2
  beta~  Pi  /            beta~  (1 + %1)       beta~  (1 + %1)
               -infinity

                            5   2
           2 (-_t0 + alpha~)  %1
         - ---------------------- d_t0
                   3         3
              beta~  (1 + %1)

           /              infinity                         \
           |             /                        4        |
           |    36      |          (-_t0 + alpha~)  %1     |
         - |----------  |          ------------------- d_t0|
           |     5   4  |                          2       |
           |beta~  Pi  /             beta~ (1 + %1)        |
           \             -infinity                         /

              -_t0 + alpha~
  %1 := exp(- -------------)
                  beta~

> plot(subs(valnum,pdf),x=-10..10);

> 
