> 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));
> cdf:=subs(u=(x-a)/(b-a),CDF(beta_,u));
> mu_:=a+(b-a)*Mean(beta_);
> var_:=(b-a)^2*Variance(beta_);
> 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=1,t=3,a=-2,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,var_));

                    /-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)


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


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

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

                                     2
               -a mu + b a + var + mu  - mu b
        t~ = - ------------------------------}
                            var


                             qdf := FAIL


                  /  _Z - a\r~
  qdf2 := RootOf(-|- ------|
                  \  -b + a/

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

        (-t~ + r~ + 1)

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

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

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

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


                            -0.1250000000


                             0.1250000000


                             0.9375000000


  [0.2094784941 + 0.1250000000 Psi(s), -0.1732867951

         - 0.1250000000 Psi(s) + 0.1250000000 Psi(1. + s),

        0.03125000000 s, -0.1250000000 + 0.09375000000 s]


  [-0.2697019430 + 1.500000000 diff(hypergeom([1, -1], [2], 3/4), 1),

        0.4687500000

         + 1.500000000 diff(hypergeom([1, -1], [2], 3/4), 3),

        -0.03125000000, -0.0937500000]


                             1.105572809


                            -0.6666666667


                             0.8888888889

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

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

> restart:
> with(Statistics):
> exponential_:=RandomVariable(Exponential(1/lambda)):
> assume(u>0);
> pdf:=subs(u=x-'gamma',PDF(exponential_,u));
> cdf:=subs(u=x-'gamma',CDF(exponential_,u));
> mu_:='gamma'+Mean(exponential_);
> var_:=Variance(exponential_);
> qdf:='gamma'+Quantile(exponential_,p);
> qdf2:=solve(cdf=p,x);
> [factor(diff(pdf,lambda))];
> [factor(diff(cdf,lambda))];

                pdf := lambda exp(-(x - gamma) lambda)


                 cdf := 1 - exp(-(x - gamma) lambda)


                                         1
                        mu_ := gamma + ------
                                       lambda


                                      1
                           var_ := -------
                                         2
                                   lambda


                                      ln(1 - p)
                       qdf := gamma - ---------
                                       lambda


                           lambda gamma - ln(1 - p)
                   qdf2 := ------------------------
                                    lambda


      [-exp(-(x - gamma) lambda) (-1 + lambda x - lambda gamma)]


                [(x - gamma) exp(-(x - gamma) lambda)]

> restart:
> with(Statistics):
> gamma_:=RandomVariable(Gamma(1/lambda,k)):
> assume(u>0);
> pdf:=subs(u=x-_gamma,PDF(gamma_,u));
> cdf:=subs(u=x-_gamma,CDF(gamma_,u)):
> cdf2:=simplify(convert(cdf,hypergeom),symbolic);
> mu_:=_gamma+Mean(gamma_);
> var_:=Variance(gamma_);
> subs(m='mu',v='var',solve({mu_=m,var_=v},{k,lambda}));
> #qdf:=_gamma+Quantile(gamma_,p);
> #qdf2:=solve(cdf=p,x);
> map(_u->simplify(convert(_u,hypergeom),symbolic),[diff(pdf, k)/pdf,
> diff(pdf, lambda)/pdf, diff(pdf, _gamma)/pdf]);
> #dCDFdk:=collect(map(_u->simplify(convert(_u,hypergeom),symbolic),conv
> ert(diff(subs(_gamma=0,cdf2), k),hypergeometric)),x);
> #dCDFdlambda:=map(_u->factor(simplify(convert(_u,hypergeom),symbolic))
> ,diff(cdf2, lambda));
> #dCDFdgamma:=map(_u->simplify(convert(_u,hypergeom),symbolic),diff(cdf
> 2, _gamma));
> collect(expand(simplify(factor(convert(subs(_gamma=0,lambda=1,diff(cdf
> 2,k)),'StandardFunctions'))),trig),k);

                              (k - 1)
  pdf := ((x - _gamma) lambda)        exp(-(x - _gamma) lambda)

        lambda/GAMMA(k)


  cdf2 := (GAMMA(k + 2) - GAMMA(k + 1, (x - _gamma) lambda)

         - GAMMA(k + 1, (x - _gamma) lambda) k

                         k                                 k
         + k (x - _gamma)  exp(-(x - _gamma) lambda) lambda

                       k                                 k
         + (x - _gamma)  exp(-(x - _gamma) lambda) lambda )/

        GAMMA(k + 2)


                                          k
                        mu_ := _gamma + ------
                                        lambda


                                      k
                           var_ := -------
                                         2
                                   lambda


                                                       2
                       -_gamma + mu      (-_gamma + mu)
             {lambda = ------------, k = ---------------}
                           var                 var


  [ln(x - _gamma) + ln(lambda) - Psi(k),

          -k + lambda x - lambda _gamma
        - -----------------------------,
                     lambda

        -k + 1 + lambda x - lambda _gamma
        ---------------------------------]
                   x - _gamma


  k GAMMA(k + 1, x) Psi(k)   2 GAMMA(k + 1, x) Psi(k)
  ------------------------ + ------------------------
            2                          2
     (k + 1)  GAMMA(k)          (k + 1)  GAMMA(k)

           GAMMA(k + 1, x) Psi(k)   2 GAMMA(k + 1, x) ln(x)
         + ---------------------- - -----------------------
                   2                          2
            (k + 1)  k GAMMA(k)        (k + 1)  GAMMA(k)

           GAMMA(k + 1, x) ln(x)   k GAMMA(k + 1, x) ln(x)
         - --------------------- - -----------------------
                   2                         2
            (k + 1)  k GAMMA(k)       (k + 1)  GAMMA(k)

            2 GAMMA(k + 1, x)     GAMMA(k + 1, x)
         + ------------------- + -----------------
                  2                     2
           (k + 1)  k GAMMA(k)   (k + 1)  GAMMA(k)

            k
           x  x hypergeom([k + 1, k + 1], [k + 2, k + 2], -x)
         - --------------------------------------------------
                                 2
                          (k + 1)  k GAMMA(k)

                                   2                     2
             GAMMA(k + 1, x)      k  ln(x)    ln(x)     k  Psi(k)
         + -------------------- + -------- + -------- - ---------
                  2  2                   2          2          2
           (k + 1)  k  GAMMA(k)   (k + 1)    (k + 1)    (k + 1)

                                            k
            Psi(k)    2 k Psi(k)         k x  Psi(k)
         - -------- - ---------- - ------------------------
                  2           2           2
           (k + 1)     (k + 1)     (k + 1)  GAMMA(k) exp(x)

                    k                          k
                 2 x  Psi(k)                  x  Psi(k)
         - ------------------------ - --------------------------
                  2                          2
           (k + 1)  GAMMA(k) exp(x)   (k + 1)  k GAMMA(k) exp(x)

                     k                          k
                  k x  ln(x)                 2 x  ln(x)
         + ------------------------ + ------------------------
                  2                          2
           (k + 1)  GAMMA(k) exp(x)   (k + 1)  GAMMA(k) exp(x)

                     k
                    x  ln(x)            2 k ln(x)
         + -------------------------- + ---------
                  2                            2
           (k + 1)  k GAMMA(k) exp(x)   (k + 1)

                         k                          k
                      2 x                          x
         - -------------------------- - ------------------------
                  2                            2
           (k + 1)  k GAMMA(k) exp(x)   (k + 1)  GAMMA(k) exp(x)

                                   k
              2                   x                     1
         - -------- - --------------------------- - ----------
                  2          2  2                          2
           (k + 1)    (k + 1)  k  GAMMA(k) exp(x)   (k + 1)  k

              k
         - --------
                  2
           (k + 1)


> restart:
> with(Statistics):
> gumbel_:=RandomVariable(Gumbel(beta,1/alpha)):
> pdf:=PDF(gumbel_,x);
> ddf:=factor(diff(pdf,x));
> cdf:=CDF(gumbel_,x);
> mu_:=Mean(gumbel_);
> var_:=Variance(gumbel_);
> assume(v>0):
> subs(m='mu',v='var',allvalues(solve({mu_=m,var_=v},{alpha,beta})));
> qdf:=Quantile(gumbel_,p);
> qdf2:=solve(cdf=p,x);
> map(factor,[diff(pdf,alpha)/pdf,diff(pdf,beta)/pdf]);
> map(factor,[diff(cdf,alpha)/cdf,diff(cdf,beta)/cdf]);

   pdf := alpha exp(-(x - beta) alpha) exp(-exp(-(x - beta) alpha))


              2
  ddf := alpha  exp(-(x - beta) alpha) exp(-exp(-(x - beta) alpha))

        (-1 + exp(-(x - beta) alpha))


                 cdf := exp(-exp(-(x - beta) alpha))


                                       gamma
                         mu_ := beta + -----
                                       alpha


                                       2
                                     Pi
                           var_ := --------
                                          2
                                   6 alpha


                                    /                /  2\1/2\
                                    |            1/2 |Pi |   |
        /  2\1/2          /  2\1/2  |         m 6    |---|   |
    1/2 |Pi |         1/2 |Pi |     |                \v~ /   |  1/2
   6    |---|        6    |---|     |-gamma + ---------------| 6
        \v~ /             \var/     \                6       /
  {------------- = - -------------, -------------------------------
         6                 6                   /  2\1/2
                                               |Pi |
                                               |---|
                                               \v~ /

             /                 /  2\1/2\
             |             1/2 |Pi |   |
             |         mu 6    |---|   |
             |                 \var/   |  1/2
             |-gamma - ----------------| 6
             \                6        /
         = - --------------------------------}
                         /  2\1/2
                         |Pi |
                         |---|
                         \var/


                                     ln(-ln(p))
                       qdf := beta - ----------
                                       alpha


                           beta alpha - ln(-ln(p))
                   qdf2 := -----------------------
                                    alpha


  [(1 - alpha x + beta alpha + alpha exp(-(x - beta) alpha) x

         - alpha exp(-(x - beta) alpha) beta)/alpha,

        -alpha (-1 + exp(-(x - beta) alpha))]


  [(x - beta) exp(-(x - beta) alpha), -alpha exp(-(x - beta) alpha)]

> restart:
> with(Statistics):
> with(student):
> assume(u>0,sigma>0):
> lognormal_:=RandomVariable(LogNormal(mu,sigma)):
> pdf:=subs(u='x-_gamma',PDF(lognormal_,u));
> cdf:=subs(u='x-_gamma',CDF(lognormal_,u));
> cdf2:=value(changevar(ln(x)=y,Int(pdf,x=0..t)));
> mu_:='_gamma'+Mean(lognormal_);
> var_:=Variance(lognormal_);
> qdf:='_gamma'+Quantile(lognormal_,p);
> qdf2:=solve(cdf2=p,x);
> fact:=pdf:
> map(factor,[diff(pdf,mu)/fact,diff(pdf,sigma)/fact,diff(pdf,_gamma)/fa
> ct]);
> fact:=pdf:
> map(factor,[diff(cdf,mu)/fact,diff(cdf,sigma)/fact,diff(cdf,_gamma)/fa
> ct]) assuming x - _gamma > exp(mu);

                                                         2
                       1/2          (ln(x - _gamma) - mu)
                      2    exp(-1/2 ----------------------)
                                                 2
                                           sigma~
           pdf := 1/2 -------------------------------------
                                                  1/2
                            (x - _gamma) sigma~ Pi


         {                                          1/2
  cdf := {                   (mu - ln(x - _gamma)) 2
         { 1/2 - 1/2 erf(1/2 --------------------------) ,
         {                             sigma~

        x - _gamma < exp(mu)

                                                 1/2
                          (ln(x - _gamma) - mu) 2
        1/2 + 1/2 erf(1/2 --------------------------) , otherwise
                                    sigma~


                       1/2
                      2    (mu - ln(-_gamma))
  cdf2 := 1/2 erf(1/2 -----------------------)
                              sigma~

                        1/2
                       2    (-ln(t - _gamma) + mu)
         - 1/2 erf(1/2 ---------------------------)
                                 sigma~


                                                 2
                                           sigma~
                  mu_ := _gamma + exp(mu + -------)
                                              2


                                     2             2
            var_ := exp(2 mu + sigma~ ) (exp(sigma~ ) - 1)


                             qdf := FAIL


                               qdf2 :=


     mu - ln(x - _gamma)
  [- -------------------, - (mu - ln(x - _gamma) + sigma~)
                 2
           sigma~

                                              3
        (-mu + ln(x - _gamma) + sigma~)/sigma~ ,

              2
        sigma~  + ln(x - _gamma) - mu
        -----------------------------]
                  2
            sigma~  (x - _gamma)


                      (ln(x - _gamma) - mu) (-x + _gamma)
        [-x + _gamma, -----------------------------------, -1]
                                    sigma~

> 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_:='gamma'+Mean(logistic_);
> var_:=Variance(logistic_);
> qdf:=Quantile(logistic_,p);
> qdf2:=solve(cdf=p,x);
> dpdf:=map(factor,[diff(pdf,alpha),diff(pdf,beta)]);
> factor(dpdf[2]-(x-alpha)*dpdf[1]/beta);
> dcdf:=map(factor,[diff(cdf,alpha),diff(cdf,beta)]);
> factor(dcdf[2]/dcdf[1]);

                                   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_ := gamma + 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  /
  dpdf := [------------------------------------, 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~
  dcdf := [- -----------------------------,
             /          x - alpha \2
             |1 + exp(- ---------)|  beta~
             \            beta~   /

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


                              x - alpha
                              ---------
                                beta~

> restart:
> with(Statistics):
> assume(sigma>0):
> normal_:=RandomVariable(Normal(mu,sigma)):
> pdf:=subs(u=x,PDF(normal_,u));
> cdf:=subs(u=x,CDF(normal_,u));
> cdf2:=int(pdf,x=0..t);
> mu_:=Mean(normal_);
> var_:=Variance(normal_);
> qdf:=Quantile(normal_,p);
> qdf2:=solve(cdf2=p,x);

                                                 2
                               1/2       (x - mu)
                              2    exp(- ---------)
                                                 2
                                         2 sigma~
                   pdf := 1/2 ---------------------
                                    1/2
                                  Pi    sigma~


                                       1/2
                                      2    (x - mu)
                 cdf := 1/2 + 1/2 erf(-------------)
                                        2 sigma~


                              1/2              1/2
                          mu 2                2    (t - mu)
          cdf2 := 1/2 erf(--------) + 1/2 erf(-------------)
                          2 sigma~              2 sigma~


                              mu_ := mu


                                         2
                           var_ := sigma~


                     1/2                                        1/2
   qdf := -1/2 (-mu 2    + 2 RootOf(erf(_Z) - 1 + 2 p) sigma~) 2


                               qdf2 :=

> restart:
> with(Statistics):
> assume(sigma>0):
> normal_:=RandomVariable(Normal(mu,sigma)):
> pdf:=PDF(normal_,x)/(subs(x=b,CDF(normal_,x))-subs(x=a,CDF(normal_,x))
> );
> cdf:=int(subs(x=t,pdf),t=a..x);
> mu_:=int(x*pdf,x=a..b);
> var_:=int((x-mu_)^2*pdf,x=a..b);
> qdf:=solve(cdf=p,x);
> map(_x->factor(subs(erf(sqrt(2)*(b-mu)/2/sigma)=2*PhiB-1,erf(sqrt(2)*(
> a-mu)/2/sigma)=2*PhiA-1,exp(-(b-mu)^2/2/sigma^2)=sqrt(2*Pi)*sigma*phiB
> ,exp(-(a-mu)^2/2/sigma^2)=sqrt(2*Pi)*sigma*phiA,x=mu+sigma*X,a=mu+sigm
> a*A,b=mu+sigma*B,_x)),[diff(pdf,mu),diff(pdf,sigma),diff(pdf,a),diff(p
> df,b)]);
> map(_x->factor(subs(erf(sqrt(2)*(b-mu)/2/sigma)=2*PhiB-1,erf(sqrt(2)*(
> a-mu)/2/sigma)=2*PhiA-1,exp(-(b-mu)^2/2/sigma^2)=sqrt(2*Pi)*sigma*phiB
> ,exp(-(a-mu)^2/2/sigma^2)=sqrt(2*Pi)*sigma*phiA,x=mu+sigma*X,a=mu+sigm
> a*A,b=mu+sigma*B,_x)),[diff(cdf,mu),diff(cdf,sigma),diff(cdf,a),diff(c
> df,b)]);

                                2      /
              1/2       (x - mu)     / |  1/2
  pdf := 1/2 2    exp(- ---------)  /  |Pi    sigma~
                                2  /   \
                        2 sigma~

        /         1/2                      1/2          \\
        |        2    (b - mu)            2    (a - mu) ||
        |1/2 erf(-------------) - 1/2 erf(-------------)||
        \          2 sigma~                 2 sigma~    //


                         1/2                  1/2
                        2    (a - mu)        2    (x - mu)
                   -erf(-------------) + erf(-------------)
                          2 sigma~             2 sigma~
          cdf := - ----------------------------------------
                         1/2                  1/2
                        2    (b - mu)        2    (a - mu)
                   -erf(-------------) + erf(-------------)
                          2 sigma~             2 sigma~


         /                               2    2
         |  1/2              -2 mu a + mu  + a
  mu_ := |-2    sigma~ exp(- ------------------)
         |                               2
         \                       2 sigma~

                   1/2
                  2    (a - mu)    1/2
         + mu erf(-------------) Pi
                    2 sigma~

                                         2    2
            1/2              -2 mu b + mu  + b
         + 2    sigma~ exp(- ------------------)
                                         2
                                 2 sigma~

                   1/2                \     /
                  2    (b - mu)    1/2|   / |  1/2
         - mu erf(-------------) Pi   |  /  |Pi
                    2 sigma~          | /   \
                                      /

        /      1/2                  1/2          \\
        |     2    (b - mu)        2    (a - mu) ||
        |-erf(-------------) + erf(-------------)||
        \       2 sigma~             2 sigma~    //


            /
            |          (3/2)           2       1/2
  var_ := - |-sigma~ Pi      exp(%5) %1  + %1 2    mu Pi %3
            |
            \

                            (3/2)               1/2
         + 2 %1 sigma~ %2 Pi      exp(%5) - %1 2    b Pi %3

               1/2                1/2            1/2
         - %1 2    mu Pi %4 + %1 2    a Pi %4 - 2    a %2 Pi %4

                                  2
                        2 mu a + b     1/2
         + 2 sigma~ exp(-----------) Pi
                                2
                          sigma~

                               2             2
                      1/2     a  + 2 mu a + b  + 2 mu b
         - 4 sigma~ Pi    exp(-------------------------)
                                              2
                                      2 sigma~

                    2   (3/2)            1/2
         - sigma~ %2  Pi      exp(%5) + 2    b %2 Pi %3

                                  2
                        2 mu b + a     1/2    1/2
         + 2 sigma~ exp(-----------) Pi    + 2    mu %2 Pi %4
                                2
                          sigma~

                           \
            1/2            |                   /
         - 2    mu %2 Pi %3| sigma~ exp(-%5)  /  (
                           |                 /
                           /

           2               2    3/2
        (%2  - 2 %2 %1 + %1 ) Pi   )

             1/2
            2    (a - mu)
  %1 := erf(-------------)
              2 sigma~

             1/2
            2    (b - mu)
  %2 := erf(-------------)
              2 sigma~

                       2    2      2
            2 mu b + mu  + b  + 2 a
  %3 := exp(------------------------)
                           2
                   2 sigma~

                       2    2      2
            2 mu a + mu  + a  + 2 b
  %4 := exp(------------------------)
                           2
                   2 sigma~

          2    2    2
        mu  + a  + b
  %5 := -------------
                 2
           sigma~


             /                                  1/2
             | 1/2                             2    (a - mu)
  qdf := 1/2 |2    mu + 2 RootOf(erf(_Z) - erf(-------------)
             \                                   2 sigma~

                  1/2                    1/2                  \
                 2    (b - mu)          2    (a - mu)         |  1/2
         - p erf(-------------) + p erf(-------------)) sigma~| 2
                   2 sigma~               2 sigma~            /


                    2
        1/2        X
  [1/2 2    exp(- ----) (X PhiB - X PhiA + sigma~ phiB - sigma~ phiA)
                   2

                                                              2
           /    1/2       2              2        1/2        X
          /  (Pi    sigma~  (PhiB - PhiA) ), 1/2 2    exp(- ----) (
         /                                                   2

                        2         2
        -PhiB + PhiA + X  PhiB - X  PhiA + sigma~ phiB B

                            /    1/2       2              2
         - sigma~ phiA A)  /  (Pi    sigma~  (PhiB - PhiA) ),
                          /

                       2
                      X     1/2
               exp(- ----) 2    phiA
                      2
        1/2 ---------------------------,
              1/2                     2
            Pi    sigma~ (PhiB - PhiA)

                        2
                       X     1/2
                exp(- ----) 2    phiB
                       2
        -1/2 ---------------------------]
               1/2                     2
             Pi    sigma~ (PhiB - PhiA)


       /                                        2
       |                1/2         1/2        X
  [1/2 |2 phiA sigma~ Pi    PhiB - 2    exp(- ----) PhiB
       \                                       2

                        2
            1/2        X                      1/2
         + 2    exp(- ----) PhiA - 2 sigma~ Pi    PhiA phiB
                       2

                    1/2                      1/2
         + sigma~ Pi    phiB - phiA sigma~ Pi

                             1/2
                    1/2     2    X
         + sigma~ Pi    erf(------) phiB
                              2

                             1/2        \
                    1/2     2    X      |   /    1/2
         - sigma~ Pi    erf(------) phiA|  /  (Pi    sigma~
                              2         / /

                              /
                     2        |                   1/2
        (PhiB - PhiA) ), -1/2 |-2 phiA A sigma~ Pi    PhiB
                              \

                          2                         2
            1/2          X            1/2          X
         + 2    X exp(- ----) PhiB - 2    X exp(- ----) PhiA
                         2                         2

                      1/2                        1/2
         + 2 sigma~ Pi    PhiA phiB B - sigma~ Pi    phiB B

                                                   1/2
                           1/2            1/2     2    X
         + phiA A sigma~ Pi    - sigma~ Pi    erf(------) phiB B
                                                    2

                             1/2          \
                    1/2     2    X        |   /    1/2
         + sigma~ Pi    erf(------) phiA A|  /  (Pi    sigma~
                              2           / /

                                  /                   1/2   \
                                  |                  2    X |
                             phiA |-2 PhiB + 1 + erf(------)|
                     2            \                    2    /
        (PhiB - PhiA) ), 1/2 --------------------------------,
                                                   2
                                      (PhiB - PhiA)

             /                   1/2   \
             |                  2    X |
             |-2 PhiA + 1 + erf(------)| phiB
             \                    2    /
        -1/2 --------------------------------]
                                   2
                      (PhiB - PhiA)

> restart:
> with(Statistics):
> assume(u>0,nu>0):
> student_:=RandomVariable(StudentT(nu)):
> pdf:=subs(u=x-mu,PDF(student_,u));
> cdf:=subs(u=x-mu,CDF(student_,u)):
> cdf2:=simplify(convert(subs(mu=0,(cdf - 1)*Beta(1/2,nu/2)*sqrt(nu)),
> GAMMA)) assuming nu > 1:
> mu_:=mu+Mean(student_);
> var_:=Variance(student_);
> factor(diff((1+(x-mu)^2/nu)^(-1/2-1/2*nu),x));
> map(factor,[diff((1+(x-mu)^2/nu)^(-1/2-1/2*nu),mu),diff(pdf,nu)]);
> #qdf:=Quantile(student_,p):
> #qdf2:=solve(cdf=p,x):

                                          nu~
                              GAMMA(1/2 + ---)
                                           2
       pdf := -------------------------------------------------
                                                    /      nu~\
                                                    |1/2 + ---|
                                                    \       2 /
                                     /            2\
                      1/2       nu~  |    (x - mu) |
              (Pi nu~)    GAMMA(---) |1 + ---------|
                                 2   \       nu~   /


                          /{ undefined        nu~ <= 1 \
              mu_ := mu + |{                           |
                          \{     0            otherwise/


                         { undefined        nu~ <= 2
                         {
                 var_ := {   nu~
                         { --------         otherwise
                         { -2 + nu~


                                /        nu~\
                                |- 1/2 - ---|
                                \         2 /
       /       2              2\
       |nu~ + x  - 2 x mu + mu |
       |-----------------------|              (nu~ + 1) (x - mu)
       \          nu~          /
     - ---------------------------------------------------------
                               2              2
                        nu~ + x  - 2 x mu + mu


        /        nu~\
        |- 1/2 - ---|
        \         2 /
   /%1 \
   |---|              (nu~ + 1) (x - mu)
   \nu~/                                                  nu~
  [-------------------------------------, 1/2 GAMMA(1/2 + ---) Pi nu~
                    %1                                     2

        /          nu~                  nu~   2
        |Psi(1/2 + ---) nu~ + Psi(1/2 + ---) x
        \           2                    2

                       nu~                   nu~    2
         - 2 Psi(1/2 + ---) x mu + Psi(1/2 + ---) mu  - 1
                        2                     2

               nu~            nu~   2         nu~
         - Psi(---) nu~ - Psi(---) x  + 2 Psi(---) x mu
                2              2               2

               nu~    2      %1            %1    2        %1
         - Psi(---) mu  - ln(---) nu~ - ln(---) x  + 2 ln(---) x mu
                2            nu~           nu~            nu~

                                                /
                                                |
                                                |
              %1     2    2              2\   / |        3/2
         - ln(---) mu  + x  - 2 x mu + mu |  /  |(Pi nu~)
              nu~                         / /   \

                        /      nu~\   \
                        |1/2 + ---|   |
                        \       2 /   |
              nu~  /%1 \              |
        GAMMA(---) |---|            %1|]
               2   \nu~/              /

               2              2
  %1 := nu~ + x  - 2 x mu + mu

> restart:
> with(Statistics):
> assume(a<m,m<b,a<b,x>a,x<b):
> triangular_:=RandomVariable(Triangular(a,b,m)):
> pdf:=subs(u=x,PDF(triangular_,u));
> ddf:=diff(pdf,x);
> cdf:=subs(u=x,CDF(triangular_,u));
> mu_:=Mean(triangular_);
> var_:=Variance(triangular_);
> qdf:=Quantile(triangular_,p);
> map(factor,[diff(pdf,a),diff(pdf,m),diff(pdf,b)]);
> map(factor,[diff(cdf,a),diff(cdf,m),diff(cdf,b)]);

                   {          0                  x~ < a~
                   {
                   {     2 (x~ - a~)
                   { -------------------        x~ <= m~
                   { (b~ - a~) (m~ - a~)
            pdf := {
                   {     2 (b~ - x~)
                   { -------------------        x~ <= b~
                   { (b~ - a~) (b~ - m~)
                   {
                   {          0                 otherwise


                   {           2
                   {  -------------------         x~ < m~
                   {  (b~ - a~) (m~ - a~)
                   {
            ddf := {       undefined              x~ = m~
                   {
                   {            2
                   { - -------------------        m~ < x~
                   {   (b~ - a~) (b~ - m~)


                 {            0                    x~ < a~
                 {
                 {                2
                 {       (x~ - a~)
                 {   -------------------          x~ <= m~
                 {   (b~ - a~) (m~ - a~)
          cdf := {
                 {                  2
                 {         (b~ - x~)
                 { 1 - -------------------        x~ <= b~
                 {     (b~ - a~) (b~ - m~)
                 {
                 {            1                   otherwise


                              a~     b~     m~
                      mu_ := ---- + ---- + ----
                              3      3      3


                 2          2          2
  var_ := 1/18 a~  + 1/18 b~  + 1/18 m~  - 1/18 a~ b~ - 1/18 a~ m~

         - 1/18 b~ m~


         {                                1/2               m~ - a~
         {    a~ + (p (b~ - a~) (m~ - a~))              p < -------
  qdf := {                                                  b~ - a~
         {
         {                                   1/2
         { b~ - ((1 - p) (b~ - a~) (b~ - m~))            otherwise


   {               2
   { 2 (-b~ m~ + a~  + x~ m~ - 2 x~ a~ + x~ b~)
   { ------------------------------------------        x~ <= m~
   {                    2           2
  [{          (-b~ + a~)  (-m~ + a~)                            ,
   {
   {                2 (-b~ + x~)
   {           ----------------------                  otherwise
   {                     2
   {           (-b~ + a~)  (-b~ + m~)

        {       2 (-b~ + x~)
        {  ----------------------         m~ < x~
        {                       2
        {  (-b~ + a~) (-b~ + m~)                   {
        {                                          {
        {        undefined                m~ = x~, {
        {                                          {
        {        2 (-x~ + a~)                      {
        { - ----------------------        x~ < m~
        {                        2
        {   (-b~ + a~) (-m~ + a~)

               2 (-x~ + a~)
        - ---------------------- , x~ <= m~
                    2
          (-b~ + a~)  (-m~ + a~)

               2
          2 (b~  - a~ m~ + x~ m~ - 2 x~ b~ + x~ a~)
        - ----------------------------------------- , otherwise]
                             2           2
                   (-b~ + a~)  (-b~ + m~)


   {
   {
  [{ (-x~ + a~) (2 b~ m~ - a~ b~ - a~ m~ - x~ m~ + 2 x~ a~ - x~ b~)/(
   {
   {

                  2           2
        (-b~ + a~)  (-m~ + a~) ) , x~ <= m~

                       2
             (-b~ + x~)
        ---------------------- , otherwise,
                  2
        (-b~ + a~)  (-b~ + m~)

        {                2
        {      (-b~ + x~)
        { ----------------------        m~ <= x~
        {                      2                  {
        { (-b~ + a~) (-b~ + m~)                   {
        {                                       , {
        {                2                        {
        {      (-x~ + a~)                         {
        { ----------------------        x~ < m~
        {                      2
        { (-b~ + a~) (-m~ + a~)

                       2
             (-x~ + a~)
        ---------------------- , x~ <= m~
                  2
        (-b~ + a~)  (-m~ + a~)

        - (-b~ + x~)

        (b~ m~ + a~ b~ - 2 a~ m~ + x~ m~ - 2 x~ b~ + x~ a~)/(

                  2           2
        (-b~ + a~)  (-b~ + m~) ) , otherwise]

> restart:
> with(Statistics):
> assume(a<b,x<b,x>a):
> uniform_:=RandomVariable(Uniform(a,b)):
> pdf:=subs(u=x,PDF(uniform_,u));
> cdf:=subs(u=x,CDF(uniform_,u));
> cdf2:=int(pdf,x=0..t);
> mu_:=Mean(uniform_);
> var_:=Variance(uniform_);
> qdf:=Quantile(uniform_,p);
> qdf2:=solve(cdf=p,x);
> [diff(pdf,a),diff(pdf,b)];
> map(factor,[diff(cdf,a),diff(cdf,b)]);

                         {    0            x~ < a~
                         {
                         {    1
                  pdf := { -------         x~ < b~
                         { b~ - a~
                         {
                         {    0           otherwise


                         {    0            x~ < a~
                         {
                         { x~ - a~
                  cdf := { -------         x~ < b~
                         { b~ - a~
                         {
                         {    1           otherwise


                                      t
                           cdf2 := -------
                                   b~ - a~


                                  a~     b~
                          mu_ := ---- + ----
                                  2      2


                                           2
                                  (b~ - a~)
                          var_ := ----------
                                      12


                       qdf := a~ + p (b~ - a~)


                       qdf2 := a~ + p b~ - p a~


                           1             1
                      [----------, - ----------]
                                2             2
                       (b~ - a~)     (b~ - a~)


                        -b~ + x~     -x~ + a~
                      [-----------, -----------]
                                 2            2
                       (-b~ + a~)   (-b~ + a~)

> restart:
> with(Statistics):
> assume(u>0,alpha>0,beta>0):
> weibull_:=RandomVariable(Weibull(alpha,beta)):
> pdf:=subs(u=x-'gamma',PDF(weibull_,u));
> ddf:=diff(pdf,x);
> cdf:=subs(u=x-'gamma',CDF(weibull_,u));
> cdf2:=int(pdf,x=0..t);
> mu_:='gamma'+Mean(weibull_);
> var_:=Variance(weibull_);
> subs(m='mu',v='var',solve({mu_=m,var_=v},{alpha,beta}));
> qdf:='gamma'+Quantile(weibull_,p);
> qdf2:=solve(cdf=p,x);
> dpdf:=[diff(pdf,alpha),diff(pdf,beta)]:
> map(factor,dpdf);
> factor(dpdf[2]/dpdf[1]);
> dcdf:=[diff(cdf,alpha),diff(cdf,beta)]:
> map(factor,dcdf);
> factor(dcdf[2]/dcdf[1]);

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


                          (-1 + beta~)
  ddf := beta~ (x - gamma)             (-1 + beta~)

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

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

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


                                  /x - gamma\beta~
                  cdf := 1 - exp(-|---------|     )
                                  \ alpha~  /


                    beta~                       (-beta~)
  cdf2 := exp(-gamma      exp(Pi beta~ I) alpha~        )

                           beta~       (-beta~)
         - exp(-(t - gamma)      alpha~        )


                                            beta~ + 1
                mu_ := gamma + alpha~ GAMMA(---------)
                                              beta~


                       /      beta~ + 2          beta~ + 1 2\
        var_ := alpha~ |GAMMA(---------) - GAMMA(---------) |
                       \        beta~              beta~    /

Warning, solutions may have been lost


                               v = var


                                               /  1  \
                                               |-----|
                                               \beta~/
                                           1
                qdf := gamma + alpha~ ln(-----)
                                         1 - p


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


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

        /     /x - gamma\beta~\   /        beta~
        |-1 + |---------|     |  /  (alpha~      alpha~), -
        \     \ alpha~  /     / /

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

         - beta~ ln(x - gamma) + beta~ ln(alpha~)

                 /x - gamma\beta~    x - gamma \   /       beta~
         + beta~ |---------|      ln(---------)|  /  alpha~     ]
                 \ alpha~  /          alpha~   / /


    /
  - |-1 - beta~ ln(x - gamma) + beta~ ln(alpha~)
    \

                 /x - gamma\beta~    x - gamma \          / /     2
         + beta~ |---------|      ln(---------)| alpha~  /  |beta~
                 \ alpha~  /          alpha~   /        /   \

        /     /x - gamma\beta~\\
        |-1 + |---------|     ||
        \     \ alpha~  /     //


     /x - gamma\beta~            /x - gamma\beta~
     |---------|      beta~ exp(-|---------|     )
     \ alpha~  /                 \ alpha~  /
  [- ---------------------------------------------,
                        alpha~

        /x - gamma\beta~    x - gamma       /x - gamma\beta~
        |---------|      ln(---------) exp(-|---------|     )]
        \ alpha~  /          alpha~         \ alpha~  /


                             x - gamma
                          ln(---------) alpha~
                              alpha~
                        - --------------------
                                 beta~

> -(-1+simplify(-beta*ln(x-gamma)+beta*ln(alpha),'symbolic')+beta*((x-ga
> mma)/alpha)^beta*ln((x-gamma)/alpha))*alpha/beta^2/(-1+((x-gamma)/alph
> a)^beta);

    /
  - |-1 - beta~ (ln(x - gamma) - ln(alpha~))
    \

                 /x - gamma\beta~    x - gamma \          / /     2
         + beta~ |---------|      ln(---------)| alpha~  /  |beta~
                 \ alpha~  /          alpha~   /        /   \

        /     /x - gamma\beta~\\
        |-1 + |---------|     ||
        \     \ alpha~  /     //

> -(-1-simplify(beta*(ln(x-gamma)-ln(alpha)),'assume =
> positive')+beta*((x-gamma)/alpha)^beta*ln((x-gamma)/alpha))*alpha/beta
> ^2/(-1+((x-gamma)/alpha)^beta);

    /
  - |-1 - beta~ (ln(x - gamma) - ln(alpha~))
    \

                 /x - gamma\beta~    x - gamma \          / /     2
         + beta~ |---------|      ln(---------)| alpha~  /  |beta~
                 \ alpha~  /          alpha~   /        /   \

        /     /x - gamma\beta~\\
        |-1 + |---------|     ||
        \     \ alpha~  /     //

> restart:
> with(Statistics):
> geometric_:=RandomVariable(Geometric(p)):
> pdf:=subs(u=n-1,ProbabilityFunction(geometric_,u));
> cdf:=simplify(subs(u=n-1,CDF(geometric_,u)));
> mu_:=factor(1+Mean(geometric_));
> var_:=Variance(geometric_);
> qdf:=simplify(1+Quantile(geometric_,q));
> qdf2:=solve(cdf=q,K);
> factor(diff(pdf, p));
> diff(cdf, p);

                     {        0                  n < 1
              pdf := {
                     {          (n - 1)
                     { p (1 - p)               otherwise


                     {          0                 n < 1
              cdf := {
                     {            floor(n)
                     { 1 - (1 - p)                1 <= n


                              mu_ := 1/p


                                    1 - p
                            var_ := -----
                                      2
                                     p


                                    ln(1 - q)
                        qdf := ceil(---------)
                                    ln(1 - p)


                               qdf2 :=


             {             0                      n < 1
             {
             {        (n - 1)
             { (1 - p)        (-1 + p n)
             { -------------------------        otherwise
             {          -1 + p


               {            0                    n < 1
               {
               {        floor(n)
               { (1 - p)         floor(n)
               { ------------------------        1 <= n
               {          1 - p

> restart:
> with(Statistics):
> poisson_:=RandomVariable(Poisson(lambda)):
> pdf:=subs(u=n,ProbabilityFunction(poisson_,u));
> expand(log(pdf)) assuming n > 0, lambda > 0;
> cdf:=simplify(subs(u=n,CDF(poisson_,u)));
> mu_:=Mean(poisson_);
> var_:=Variance(poisson_);
> qdf:=simplify(Quantile(poisson_,q));
> qdf2:=solve(cdf=q,K);
> factor(diff(pdf, lambda));
> diff(cdf, lambda);

                   {          0                    n < 0
                   {
            pdf := {       n
                   { lambda  exp(-lambda)
                   { --------------------        otherwise
                   {          n!


                    n ln(lambda) - lambda - ln(n!)


                         GAMMA(floor(n) + 1, lambda)
                  cdf := ---------------------------
                             GAMMA(floor(n) + 1)


                            mu_ := lambda


                            var_ := lambda


  qdf := RootOf(floor(_Z) - RootOf(GAMMA(_Z + 1, lambda) _Z

         + GAMMA(_Z + 1, lambda) - q GAMMA(_Z + 2)))


                               qdf2 :=


         {                 0                          n < 0
         {
         {       n
         { lambda  exp(-lambda) (n - lambda)
         { ---------------------------------        otherwise
         {             lambda n!


                            floor(n)
                      lambda         exp(-lambda)
                    - ---------------------------
                          GAMMA(floor(n) + 1)

> # 
> allvalues(qdf);

  RootOf(floor(_Z) - RootOf(GAMMA(_Z + 1, lambda) _Z

         + GAMMA(_Z + 1, lambda) - q GAMMA(_Z + 2)))

> plot(subs(lambda=10,GAMMA((n)+1,lambda)/GAMMA((n)+1)),n=0..50);

> solve(GAMMA((n)+1,lambda)/GAMMA((n)+1)=q,n);

           RootOf(-GAMMA(_Z + 1, lambda) + q GAMMA(_Z + 1))

> int(1/sqrt(2*Pi)*exp(-t^2/2),t=-infinity..x);

                                       1/2
                                      2    x
                        1/2 + 1/2 erf(------)
                                        2

> 
