/usr/share/openturns/validation/ValidLogistic.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(beta>0,beta,real,alpha,real):
> logistic_:=RandomVariable(Logistic(alpha,beta)):
> pdf:=PDF(logistic_,x);
> ddf:=factor(diff(pdf,x));
> cdf:=CDF(logistic_,x);
> cf:=subs(u=x,CharacteristicFunction(logistic_,u));
> rough:=factor(int(pdf^2,x=-infinity..infinity));
> mu_:=Mean(logistic_);
> var_:=Variance(logistic_);
> #skew_:=simplify(convert(Skewness(logistic_),GAMMA),symbolic);
> skew_:=Int((x-mu_)^3*pdf,x=-infinity..infinity)/var_^(3/2);
> #kurt_:=simplify(convert(Kurtosis(logistic_),GAMMA),symbolic);
> kurt_:=Int((x-mu_)^4*pdf,x=-infinity..infinity)/var_^2;
> 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=-1/2,beta=3/2:
> 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(solve(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_)]));
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~
exp(alpha~ x I) Pi beta~ x
cf := --------------------------
sinh(Pi beta~ x)
1
rough := -------
6 beta~
mu_ := alpha~
2 2
beta~ Pi
var_ := ----------
3
skew_ :=
infinity
/ 3 x - alpha~
1/2 | (x - alpha~) exp(----------)
3 3 | beta~
--------------- | ----------------------------- dx
2 2 3/2 | / x - alpha~ \2
(beta~ Pi ) | beta~ |1 + exp(----------)|
/ \ beta~ /
-infinity
infinity
/ 4 x - alpha~
| (x - alpha~) exp(----------)
9 | beta~
kurt_ := ---------- | ----------------------------- dx
4 4 | / x - alpha~ \2
beta~ Pi | beta~ |1 + exp(----------)|
/ \ beta~ /
-infinity
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(----------) |
2 / x - alpha~ \3 beta~ \
beta~ |1 + exp(----------)|
\ beta~ /
x - alpha~ x - alpha~
-x + x exp(----------) + alpha~ - alpha~ exp(----------)
beta~ beta~
x - alpha~ \ / / 3
- beta~ - beta~ exp(----------)| / |beta~
beta~ / / \
/ x - alpha~ \3\
|1 + exp(----------)| |]
\ 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
0.07430698329 - 0.04059408999 I
[0.04038122117, -0.04700186026]
[-0.1310746222, -0.1310746222]
3.916658469
-0.5000000000
2.720699048
-11
0.1330935130 10
4.199999999
7.402203303
[-0.5000000000, 2.720699048]
> qdf2 := alpha-ln(-(-1+p)/p)*beta;
> diff(kurt_,alpha);
> diff(kurt_,beta);
-1 + p
qdf2 := alpha~ - ln(- ------) beta~
p
0
infinity
/ 5 4
9 | (-_t0 + alpha~) %1 (-_t0 + alpha~) %1
---------- | ------------------- - -------------------
4 4 | 3 2 2 2
beta~ Pi / beta~ (1 + %1) beta~ (1 + %1)
-infinity
5 2
2 (-_t0 + alpha~) %1
- ---------------------- d_t0
3 3
beta~ (1 + %1)
/ infinity \
| / 4 |
| 36 | (-_t0 + alpha~) %1 |
- |---------- | ------------------- d_t0|
| 5 4 | 2 |
|beta~ Pi / beta~ (1 + %1) |
\ -infinity /
-_t0 + alpha~
%1 := exp(- -------------)
beta~
> plot(subs(valnum,pdf),x=-10..10);
>
|