> restart:
> with(Statistics):
> poisson_:=RandomVariable(Poisson(lambda)):
> pdf:=subs(u=n,ProbabilityFunction(poisson_,u));
> cdf:=simplify(subs(u=n,CDF(poisson_,u)));
> mu_:=Mean(poisson_);
> var_:=Variance(poisson_);
> skew_:=simplify(convert(Skewness(poisson_),GAMMA),symbolic);
> kurt_:=simplify(convert(Kurtosis(poisson_),GAMMA),symbolic);
> qdf:=simplify(Quantile(poisson_,q));
> qdf2:=solve(cdf=q,K);
> pdfgr:=[factor(diff(pdf, lambda))];
> cdfgr:=[diff(cdf, lambda)];
> valnum:=lambda=10:
> evalf(subs(valnum,n=12,pdf));
> evalf(subs(valnum,n=12,cdf));
> evalf(subs(valnum,n=12,pdfgr));
> evalf(subs(valnum,n=12,cdfgr));
> q:=evalf(Quantile(Poisson(subs(valnum,lambda)),0.95));
> evalf(subs(valnum,n=q,cdf));
> evalf(subs(valnum,mu_));
> evalf(subs(valnum,sqrt(var_)));
> evalf(subs(valnum,skew_));
> evalf(subs(valnum,kurt_));
> evalf(subs(valnum,var_));

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


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


                            mu_ := lambda


                            var_ := lambda


                                       1
                          skew_ := ---------
                                         1/2
                                   lambda


                                 3 lambda + 1
                        kurt_ := ------------
                                    lambda


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

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


                               qdf2 :=


             {                 0                          n < 0
             {
   pdfgr := [{       n                                           ]
             { lambda  exp(-lambda) (n - lambda)
             { ---------------------------------        otherwise
             {             lambda n!


                                 floor(n)
                           lambda         exp(-lambda)
               cdfgr := [- ---------------------------]
                               GAMMA(floor(n) + 1)


                            0.09478033009


                             0.7915564764


                           [0.01895606602]


                           [-0.09478033009]


                               q := 15.


                             0.9512595968


                                 10.


                             3.162277660


                             0.3162277660


                             3.100000000


                                 10.

> fsolve(subs(lambda=0.2,GAMMA(n+1,lambda)/GAMMA(n+1))=0.95,n);# 

                             0.5754590269

> plot(map(_x->subs(lambda=30,_x),[GAMMA(floor(n)+1,lambda)/GAMMA(floor(
> n)+1),CDF(Gamma(1,lambda),n)]),n=0..100);

> cdf:=convert(CDF(Gamma(1,lambda),n),GAMMA);
> cdf2:=1-GAMMA(lambda,n)/GAMMA(lambda);
> cdf3:=GAMMA(n,lambda)/GAMMA(n);
> evalf(subs(lambda=100,n=100,[cdf,cdf2,cdf3]));

         {
         {
  cdf := {
         { 0 , n < 0
         {

                   /
                   |
                   |
        exp(- n/2) |-exp(n/2) (-GAMMA(2 + lambda)
                   \

         + GAMMA(1 + lambda, n) + GAMMA(1 + lambda, n) lambda)

            /lambda    \
            |------ - 1|
            \  2       /            lambda      lambda
         + n             WhittakerM(------ + 1, ------ + 1/2, n) +
                                      2           2

         /lambda    \                                               \
         |------ - 1|                                               |
         \  2       /            lambda      lambda                 |
        n             WhittakerM(------ + 1, ------ + 1/2, n) lambda|
                                   2           2                    /

        /(GAMMA(lambda) lambda (1 + lambda)) , 0 <= n


                                 GAMMA(lambda, n)
                     cdf2 := 1 - ----------------
                                  GAMMA(lambda)


                               GAMMA(n, lambda)
                       cdf3 := ----------------
                                   GAMMA(n)


              [0.5132987985, 0.5132987982, 0.4867012018]

> with(plots):
> series(diff(GAMMA(n,x)/GAMMA(n),x),n);

                                                        /
                                                        |
                                                        |
     exp(-x)     /  ln(x) exp(-x)   exp(-x) gamma\  2   |
   - ------- n + |- ------------- - -------------| n  + |
        x        \        x               x      /      \

                                      /  2        2\
                                      |Pi    gamma |
                  2           exp(-x) |--- + ------|
             ln(x)  exp(-x)           \12      2   /
        -1/2 -------------- + ----------------------
                   x                    x

                                        \      /
                                        |      |
                                        |      |
           exp(-x) (ln(x) + gamma) gamma|  3   |
         - -----------------------------| n  + |
                         x              /      \

                                /  2        2\
                                |Pi    gamma |
        exp(-x) (ln(x) + gamma) |--- + ------|
                                \12      2   /
        --------------------------------------
                          x

                   /                 2              3\
                   |               Pi  gamma   gamma |
           exp(-x) |-1/3 Zeta(3) - --------- - ------|
                   \                  12         6   /
         + -------------------------------------------
                                x

                    3
               ln(x)  exp(-x)
         - 1/6 -------------- - 1/12
                     x

                                                                  \
                                                                  |
                        2     2          2                        |
        exp(-x) (6 ln(x)  - Pi  + 6 gamma  + 12 gamma ln(x)) gamma|
        ----------------------------------------------------------|
                                    x                             /

             /
             |
             |
         4   |
        n  + |
             \

                                /                 2              3\
                                |               Pi  gamma   gamma |
        exp(-x) (ln(x) + gamma) |-1/3 Zeta(3) - --------- - ------|
                                \                  12         6   /
        -----------------------------------------------------------
                                     x

                   /  4                         2      2        4\
                   |Pi                        Pi  gamma    gamma |
           exp(-x) |--- + 1/3 Zeta(3) gamma + ---------- + ------|
                   \160                           24         24  /
         + -------------------------------------------------------
                                      x

                     4
                ln(x)  exp(-x)
         - 1/24 -------------- + 1/12 exp(-x)
                      x

                                                     /  2        2\
                2     2          2                   |Pi    gamma |
        (6 ln(x)  - Pi  + 6 gamma  + 12 gamma ln(x)) |--- + ------|/x
                                                     \12      2   /

                                  2                2     2
         - 1/12 exp(-x) (-ln(x) Pi  + 6 ln(x) gamma  - Pi  gamma

                                                                    \
                                                                    |
                                                                    |
                  3                      3                2         |
         + 2 gamma  + 4 Zeta(3) + 2 ln(x)  + 6 gamma ln(x) ) gamma/x|
                                                                    /

         5      6
        n  + O(n )

> plot(log(subs(n=2,GAMMA(n+1,x)-x*GAMMA(n,x)))/x^(1/4),x=1..10);

> for i from 0 to 99 do
>   level:=(0.5+i)/100.0;
>   q:=evalf(Quantile(Poisson(subs(valnum,lambda)),level));
> print([level,q]);
> od:
> 

                         [0.005000000000, 3.]


                         [0.01500000000, 4.]


                         [0.02500000000, 4.]


                         [0.03500000000, 5.]


                         [0.04500000000, 5.]


                         [0.05500000000, 5.]


                         [0.06500000000, 5.]


                         [0.07500000000, 6.]


                         [0.08500000000, 6.]


                         [0.09500000000, 6.]


                          [0.1050000000, 6.]


                          [0.1150000000, 6.]


                          [0.1250000000, 6.]


                          [0.1350000000, 7.]


                          [0.1450000000, 7.]


                          [0.1550000000, 7.]


                          [0.1650000000, 7.]


                          [0.1750000000, 7.]


                          [0.1850000000, 7.]


                          [0.1950000000, 7.]


                          [0.2050000000, 7.]


                          [0.2150000000, 7.]


                          [0.2250000000, 8.]


                          [0.2350000000, 8.]


                          [0.2450000000, 8.]


                          [0.2550000000, 8.]


                          [0.2650000000, 8.]


                          [0.2750000000, 8.]


                          [0.2850000000, 8.]


                          [0.2950000000, 8.]


                          [0.3050000000, 8.]


                          [0.3150000000, 8.]


                          [0.3250000000, 8.]


                          [0.3350000000, 9.]


                          [0.3450000000, 9.]


                          [0.3550000000, 9.]


                          [0.3650000000, 9.]


                          [0.3750000000, 9.]


                          [0.3850000000, 9.]


                          [0.3950000000, 9.]


                          [0.4050000000, 9.]


                          [0.4150000000, 9.]


                          [0.4250000000, 9.]


                          [0.4350000000, 9.]


                          [0.4450000000, 9.]


                          [0.4550000000, 9.]


                         [0.4650000000, 10.]


                         [0.4750000000, 10.]


                         [0.4850000000, 10.]


                         [0.4950000000, 10.]


                         [0.5050000000, 10.]


                         [0.5150000000, 10.]


                         [0.5250000000, 10.]


                         [0.5350000000, 10.]


                         [0.5450000000, 10.]


                         [0.5550000000, 10.]


                         [0.5650000000, 10.]


                         [0.5750000000, 10.]


                         [0.5850000000, 11.]


                         [0.5950000000, 11.]


                         [0.6050000000, 11.]


                         [0.6150000000, 11.]


                         [0.6250000000, 11.]


                         [0.6350000000, 11.]


                         [0.6450000000, 11.]


                         [0.6550000000, 11.]


                         [0.6650000000, 11.]


                         [0.6750000000, 11.]


                         [0.6850000000, 11.]


                         [0.6950000000, 11.]


                         [0.7050000000, 12.]


                         [0.7150000000, 12.]


                         [0.7250000000, 12.]


                         [0.7350000000, 12.]


                         [0.7450000000, 12.]


                         [0.7550000000, 12.]


                         [0.7650000000, 12.]


                         [0.7750000000, 12.]


                         [0.7850000000, 12.]


                         [0.7950000000, 13.]


                         [0.8050000000, 13.]


                         [0.8150000000, 13.]


                         [0.8250000000, 13.]


                         [0.8350000000, 13.]


                         [0.8450000000, 13.]


                         [0.8550000000, 13.]


                         [0.8650000000, 14.]


                         [0.8750000000, 14.]


                         [0.8850000000, 14.]


                         [0.8950000000, 14.]


                         [0.9050000000, 14.]


                         [0.9150000000, 14.]


                         [0.9250000000, 15.]


                         [0.9350000000, 15.]


                         [0.9450000000, 15.]


                         [0.9550000000, 16.]


                         [0.9650000000, 16.]


                         [0.9750000000, 17.]


                         [0.9850000000, 17.]


                         [0.9950000000, 19.]

> 
