/usr/share/openturns/validation/ValidRayleigh.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):
> assume(u>0,sigma>0):
> rayleigh_:=RandomVariable(Rayleigh(sigma)):
> pdf:=subs(u=x-_gamma,PDF(rayleigh_,u));
> ddf:=factor(diff(pdf,x));
> cdf:=subs(u=x-_gamma,CDF(rayleigh_,u));
> cf:=CharacteristicFunction(rayleigh_,x)*exp(I*x*_gamma);
> mu_:=_gamma+Mean(rayleigh_);
> evalf[25](mu_);
> var_:=Variance(rayleigh_);
> evalf[25](sqrt(var_));
> evalf[25](var_);
> skew_:=simplify((Skewness(rayleigh_)));
> evalf[25](skew_);
> kurt_:=simplify((Kurtosis(rayleigh_)));
> evalf[25](kurt_);
> qdf:=_gamma+Quantile(rayleigh_,p);
> pdfgr:=map(factor,[diff(pdf,sigma),diff(pdf,_gamma)]);
> cdfgr:=map(factor,[diff(cdf,sigma),diff(cdf,_gamma)]);
> valnum:=sigma=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,pdfgr)));
> evalf(subs(valnum,x=1,cdfgr));
> evalf(subs(valnum,p=0.95,qdf));
> evalf(subs(valnum,mu_));
> evalf(subs(valnum,sqrt(var_)));
> evalf(subs(valnum,skew_));
> evalf(subs(valnum,kurt_));
> evalf(subs(valnum,var_));
2
(x - _gamma)
(x - _gamma) exp(- -------------)
2
2 sigma~
pdf := ---------------------------------
2
sigma~
2
(x - _gamma)
ddf := - exp(- -------------) (-_gamma + x - sigma~)
2
2 sigma~
/ 4
(-_gamma + x + sigma~) / sigma~
/
2
(x - _gamma)
cdf := 1 - exp(- -------------)
2
2 sigma~
/ / 1 \(- k/2) k\
|infinity GAMMA(1 + k/2) |---------| (x I) |
| ----- | 2| |
| \ \2 sigma~ / |
cf := | ) ----------------------------------------|
| / k! |
| ----- |
\ k = 0 /
exp(x _gamma I)
1/2 1/2
sigma~ 2 Pi
mu_ := _gamma + -----------------
2
_gamma + 1.253314137315500251207882 sigma~
/ Pi \ 2
var_ := |2 - ----| sigma~
\ 2 /
0.6551363775620335530939357 sigma~
2
0.429203673205103380768678 sigma~
1/2 1/2
2 2 Pi (-3 + Pi)
skew_ := - -----------------------
1/2
(-4 + Pi) (8 - 2 Pi)
0.6311106578189371381918970
2
-32 + 3 Pi
kurt_ := - -----------
2
(-4 + Pi)
3.245089300687638062848662
2 1/2
qdf := _gamma + (-2 sigma~ ln(1 - p))
2
(x - _gamma)
pdfgr := [(x - _gamma) exp(- -------------)
2
2 sigma~
2 2 2 / 5
(-2 sigma~ + x - 2 x _gamma + _gamma ) / sigma~ ,
/
2
(x - _gamma)
exp(- -------------) (-_gamma + x - sigma~)
2
2 sigma~
/ 4
(-_gamma + x + sigma~) / sigma~ ]
/
2
2 (x - _gamma)
(x - _gamma) exp(- -------------)
2
2 sigma~
cdfgr := [- ----------------------------------,
3
sigma~
2
(x - _gamma)
(x - _gamma) exp(- -------------)
2
2 sigma~
- ---------------------------------]
2
sigma~
valnum := sigma~ = 2.5, _gamma = -0.5
0.08553166965
0.2004648507
0.1647297886
-0.1575153039 + 0.2429216564 I
[-0.1315049421, -0.08553166965]
[-0.1202789104, -0.2004648507]
5.619367077
2.633285342
1.637840944
0.6311106598
3.245089294
2.682522956
> cf2:=proc(x, sigma, _gamma)
> local t;
> t := sigma * x / sqrt(2);
> (1 - t * (2 * dawson(t) - I * exp(-t * t) * sqrt(Pi))) * exp(I *
> _gamma * x)
> end:
> evalf(cf2(1.0, 2.5, -0.5));
-0.1575153040 + 0.2429216564 I
>
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