> restart:
> with(Statistics):
> assume(beta>0):
> logistic_:=RandomVariable(Logistic(alpha,beta)):
> pdf:=PDF(logistic_,x);
> ddf:=factor(diff(pdf,x));
> cdf:=CDF(logistic_,x);
> mu_:=Mean(logistic_);
> var_:=Variance(logistic_);
> 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=-0.5,beta=1.5:
> 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,pdfgr)));
> evalf(subs(valnum,x=1,cdfgr));
> evalf(solve(subs(valnum,cdf)=0.95,x));
> evalf(subs(valnum,mu_));
> 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~


                             mu_ := alpha


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


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

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

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


                   [0.04038122117, -0.04700186026]


                    [-0.1310746222, -0.1310746222]


                             3.916658468


                                 -0.5


                             7.402203303


                         [-0.5, 2.720699048]

