/usr/share/openturns/validation/ValidGamma.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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> 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);
> cf:=subs(u=x,exp(-I*_gamma)*CharacteristicFunction(gamma_,u));
> mu_:=_gamma+Mean(gamma_);
> var_:=Variance(gamma_);
> skew_:=simplify(convert(Skewness(gamma_),GAMMA),symbolic);
> kurt_:=simplify(convert(Kurtosis(gamma_),GAMMA),symbolic);
> sol:=subs(m='mu',v='var',solve({mu_=m,var_=v},{k,lambda}));
> #qdf:=_gamma+Quantile(gamma_,p);
> #qdf2:=solve(cdf=p,x);
> pdfgr:=map(_u->simplify(convert(_u,hypergeom),symbolic),[diff(pdf,
> k)/pdf, diff(pdf, lambda)/pdf, diff(pdf, _gamma)/pdf]);
> cdfgr:=map(_u->simplify(convert(_u,hypergeom),symbolic),[diff(cdf, k),
> diff(cdf, lambda), diff(cdf, _gamma)]);
> #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);
> valnum:=k=1.5,lambda=2.5,_gamma=-0.5;
> 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*pdf,pdfgr)));
> evalf(subs(valnum,x=1,cdfgr));
> evalf(fsolve(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,[k,lambda]));
(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
+ exp(-(x - _gamma) lambda) lambda (x - _gamma)
k k
+ k exp(-(x - _gamma) lambda) lambda (x - _gamma) )/
GAMMA(k + 2)
/ x I \(-k)
cf := exp(-I _gamma) |1 - ------|
\ lambda/
k
mu_ := _gamma + ------
lambda
k
var_ := -------
2
lambda
2
skew_ := ----
1/2
k
3 (k + 2)
kurt_ := ---------
k
2
-_gamma + mu (-_gamma + mu)
sol := {lambda = ------------, k = ---------------}
var var
pdfgr := [ln(x - _gamma) + ln(lambda) - Psi(k),
k - lambda x + lambda _gamma
----------------------------,
lambda
k - 1 - lambda x + lambda _gamma
- --------------------------------]
x - _gamma
(k + 2) (k + 1)
cdfgr := [(lambda (x - _gamma)
hypergeom([k + 2, k + 2], [k + 3, k + 3], %1) _gamma -
(k + 2) (k + 1)
lambda (x - _gamma)
hypergeom([k + 2, k + 2], [k + 3, k + 3], %1) x
- 4 Psi(k + 2) GAMMA(k + 2)
k k
+ 4 k exp(%1) lambda (x - _gamma)
k k
- 8 exp(%1) Psi(k + 2) lambda (x - _gamma) k
k k 3
- exp(%1) Psi(k + 2) lambda (x - _gamma) k
k k 2
- 5 exp(%1) Psi(k + 2) lambda (x - _gamma) k
k k 2
+ 5 exp(%1) lambda (x - _gamma) ln(x - _gamma) k
k k
+ 8 exp(%1) lambda (x - _gamma) ln(x - _gamma) k
k k 3
+ exp(%1) lambda ln(lambda) (x - _gamma) k
k k 3
+ exp(%1) lambda (x - _gamma) ln(x - _gamma) k
k k 2
+ 5 exp(%1) lambda ln(lambda) (x - _gamma) k
k k
+ 8 exp(%1) lambda ln(lambda) (x - _gamma) k
k k 3
+ 4 exp(%1) lambda (x - _gamma) + %2 Psi(k + 2) k
2 3
+ 5 %2 Psi(k + 2) k + 8 %2 Psi(k + 2) k - %2 ln(lambda) k
2
- 5 %2 ln(lambda) k - 8 %2 ln(lambda) k
3 2
- %2 ln(x - _gamma) k - 5 %2 ln(x - _gamma) k
2
- 8 %2 ln(x - _gamma) k - Psi(k + 2) GAMMA(k + 2) k
- 4 Psi(k + 2) GAMMA(k + 2) k
2
+ ln(x - _gamma) GAMMA(k + 2) k
+ 4 ln(x - _gamma) GAMMA(k + 2) k
2
+ ln(lambda) GAMMA(k + 2) k + 4 ln(lambda) GAMMA(k + 2) k
k k
+ 4 exp(%1) lambda ln(lambda) (x - _gamma)
k k
- 4 exp(%1) Psi(k + 2) lambda (x - _gamma)
k k
+ 4 exp(%1) lambda (x - _gamma) ln(x - _gamma)
k k 2
+ exp(%1) lambda (x - _gamma) k
+ 4 ln(lambda) GAMMA(k + 2) + 4 ln(x - _gamma) GAMMA(k + 2)
+ 4 %2 Psi(k + 2) - 4 %2 ln(x - _gamma) - 4 %2 ln(lambda))
/ 2 (k + 1) k
/ ((k + 2) GAMMA(k + 2)), (-lambda (x - _gamma) x
/
(k + 1) k
- lambda (x - _gamma) k x
(k + 1) k
+ lambda (x - _gamma) _gamma
(k + 1) k
+ lambda (x - _gamma) k _gamma
(k + 1) (k + 1)
+ lambda (x - _gamma) k
(k + 1) (k + 1)
+ lambda (x - _gamma)
k k k k 2
+ lambda (x - _gamma) k + lambda (x - _gamma) k )
exp(%1)/(GAMMA(k + 2) lambda), - exp(%1) (
(k + 1) k
-lambda (x - _gamma) x
(k + 1) k
- lambda (x - _gamma) k x
(k + 1) k
+ lambda (x - _gamma) _gamma
(k + 1) k
+ lambda (x - _gamma) k _gamma
(k + 1) (k + 1)
+ lambda (x - _gamma) k
(k + 1) (k + 1)
+ lambda (x - _gamma)
k k k k 2
+ lambda (x - _gamma) k + lambda (x - _gamma) k )/(
(x - _gamma) GAMMA(k + 2))]
%1 := -(x - _gamma) lambda
%2 := GAMMA(k + 1, (x - _gamma) lambda)
2 k 2 GAMMA(k + 1, x)
- -------- - -------- + -------------------
2 2 2
(k + 1) (k + 1) (k + 1) k GAMMA(k)
k k
2 x x
- -------------------------- - ------------------------
2 2
(k + 1) k GAMMA(k) exp(x) (k + 1) GAMMA(k) exp(x)
k GAMMA(k + 1, x) ln(x) 2 GAMMA(k + 1, x) ln(x)
- ----------------------- - -----------------------
2 2
(k + 1) GAMMA(k) (k + 1) GAMMA(k)
GAMMA(k + 1, x) Psi(k) 2 GAMMA(k + 1, x) Psi(k)
+ ---------------------- + ------------------------
2 2
(k + 1) k GAMMA(k) (k + 1) GAMMA(k)
GAMMA(k + 1, x) ln(x) k GAMMA(k + 1, x) Psi(k)
- --------------------- + ------------------------
2 2
(k + 1) k GAMMA(k) (k + 1) GAMMA(k)
k
x 1 ln(x)
- --------------------------- - ---------- + --------
2 2 2 2
(k + 1) k GAMMA(k) exp(x) (k + 1) k (k + 1)
2
k Psi(k) GAMMA(k + 1, x) GAMMA(k + 1, x)
- --------- + ----------------- + --------------------
2 2 2 2
(k + 1) (k + 1) GAMMA(k) (k + 1) k GAMMA(k)
2
Psi(k) 2 k Psi(k) k ln(x) 2 k ln(x)
- -------- - ---------- + -------- + ---------
2 2 2 2
(k + 1) (k + 1) (k + 1) (k + 1)
k k
k x ln(x) x ln(x)
+ ------------------------ + --------------------------
2 2
(k + 1) GAMMA(k) exp(x) (k + 1) k GAMMA(k) exp(x)
k k
2 x ln(x) 2 x Psi(k)
+ ------------------------ - ------------------------
2 2
(k + 1) GAMMA(k) exp(x) (k + 1) GAMMA(k) exp(x)
k
x x hypergeom([k + 1, k + 1], [k + 2, k + 2], -x)
- --------------------------------------------------
2
(k + 1) k GAMMA(k)
k k
k x Psi(k) x Psi(k)
- ------------------------ - --------------------------
2 2
(k + 1) GAMMA(k) exp(x) (k + 1) k GAMMA(k) exp(x)
valnum := k = 1.5, lambda = 2.5, _gamma = -0.5
ddf
0.1284713816
0.9424415478
0.4289502207 + 0.7851195123 I
[0.1651198816, -0.1156242435, 0.2783546603]
[-0.08756249102, 0.07708282900, -0.1284713816]
1.062945580
0.1000000000
0.4898979484
1.632993162
6.999999999
0.2400000000
[0.1000000000, 0.4898979484]
[1.5, 2.5]
> restart:assume(k,integer,k>2):
> eq:=F(k+1)=F(k)-x^k*exp(-x)/k/GAMMA(k);
> collect(expand(solve(eq,F(k+1))),F(k));
k~
x exp(-x)
eq := F(k~ + 1) = F(k~) - ------------
k~ GAMMA(k~)
k~
x
F(k~) - -------------------
k~ GAMMA(k~) exp(x)
> map(factor,collect(expand(solve(subs(k=k-1,eq),F(k-1))),F(k)));
k~
x
F(k~) + ------------------
GAMMA(k~) x exp(x)
> int(log(x),x);
x ln(x) - x
>
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