Integrate the following expression?

Questions
,
1

I have done numerical integration. But the code does not produce an answer.

F=(1+(-1+(1-Exp[-?*y])^-?)^?)^-2*(? *?* ?*Exp[-?*y]*
(1-Exp[-?*y])^(-1-?)*(-1+(1-Exp[-?*y])^-?)^(-1+?));

Integrate[Exp[t*y]*F,{y,0, ?}]

2.nb (4.34 KB)

2

For numerical integration use: NIntegrate.

3

Mariusz: I think there are two issues: (1) Despite calling for “numerical” integration, I think a symbolic integration is what is wanted, and (2) The posting is a request as opposed to a question and I can’t tell if that is a language issue or a point of view issue.

It appears that what is wanted is a symbolic answer that will give a moment generating function in terms of \psi, \phi, and \lambda.

The density function for the random variable Y is

f = (1 + (-1 + (1 -  Exp[-\[Delta]*y])^-\[Psi])^\[Phi])^-2*(\[Delta] *\[Phi]* \[Psi]*
     Exp[-\[Delta]*y]*(1 -Exp[-\[Delta]*y])^(-1 - \[Psi])*(-1 + (1 - Exp[-\[Delta]*y])^-\[Psi])^(-1 + \[Phi]));
f=\frac{\delta \psi \phi \exp (-\delta y) (1-\exp (-\delta y))^{-\psi -1} \left((1-\exp (-\delta y))^{-\psi }-1\right)^{\phi -1}}{\left(\left((1-\exp (-\delta y))^{-\psi }-1\right)^{\phi }+1\right)^2}

If one sets \phi \to 1, then with there seems to be a symbolic answer with \psi>0 and \delta>0:

mgf = Integrate[Exp[t y] f /. {\[Phi] -> 1}, {y, 0, \[Infinity]}, 
  Assumptions -> {\[Psi] > 0, \[Delta] > 0, t < \[Delta]}]
\frac{\psi \Gamma (\psi ) \Gamma \left(1-\frac{t}{\delta }\right)}{\Gamma \left(-\frac{t}{\delta }+\psi +1\right)}

I don’t know if there is a symbolic (i.e., nice closed form) result

4

Dear Sir
Why phi approaches to 1…what about if phi>0?? Thank you
Mariusz Iwaniuk, I also try NIntegrate command but it doesnot produce any result . I want mathematica must solve such complex integration .

5

func[??NumericQ, ??NumericQ, ??NumericQ, t?NumericQ] :=
NIntegrate[Exp[ty]((E^(-y ?) (1 - E^(-y ?))^(-1 - ?) (-1 + (1 - E^(-y ?))^-?)^(-1 + ?) ? ? ?)/(1 + (-1 +
(1 - E^(-y ?))^-?)^?)^2), {y, 0, ?}, Method → “AdaptiveQuasiMonteCarlo”,
Exclusions → {(1 + (-1 + (1 - E^(-y ?))^-?)^?)^2 == 0}]

 func[2, 2, 1, 1]
 (* 2.66667 *)

 func[1/2, 2, 1/3, 1](* Probably integral is divergent !!! *)
 (* Error-> Overflow, Indeterminate, or Infinity for all sampling points *)

Regards MI.

6

Dear Mariusz
all the parameters are greater than zero. so Obviously it wont exist for any negative value of parameters . sir I didnt get the answer, what is func[[Delta]?NumericQ, [Psi]?NumericQ, [Phi]?NumericQ, t?NumericQ]
:=
Even I replace them with underscore.

7

Exist :

 func[1, -2, -2, -2]
 (* -0.0950309  *)