I am trying to evaluate
$$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some substitutions, integrating by parts to solve the integral, but it did not seem to converge.

I also tried to compute the laplace transform, inverse Fourier transform of the density using Mathematica, but it was n't able to work it out.

After having spent 6-7 hours trying to solve this I am still hopelessly stuck.

Any help would be much appreciated.

  • $\begingroup$ What did you spend "6-7 hours" at, exactly? Did you note that the RHS cannot be P(X>x) for every x, for any random variable X? $\endgroup$ – Did Sep 20 '14 at 7:48
  • $\begingroup$ @Did No I did not notice anything like that. Can you please explain more why it can not be? $\endgroup$ – Comic Book Guy Sep 20 '14 at 20:41
  • $\begingroup$ To begin with, P(X>0) might be greater than 1... $\endgroup$ – Did Sep 20 '14 at 21:35
  • $\begingroup$ @Did I think what you are getting at is that in order to compute this probability I need to rewrite the rhs as 1 - the integral with lower limit negative infinity going to x, using the complement rule. Is that correct ? $\endgroup$ – Comic Book Guy Sep 20 '14 at 21:48
  • 1
    $\begingroup$ Then you might want to mention in the question the regime you have in mind, ensuring that P(X>0) is not greater than 1. If P(X>0)<1, this means there is a Dirac delta at x=0? (FYI, I just flagged your last comment.) $\endgroup$ – Did Sep 21 '14 at 11:56

Did you try the substitution $u=t^{\alpha\kappa+1}$? As far as I see, you get then something like $k\cdot u^{\beta}\exp(-\rho u)$ ($k,\beta$ constants) as integrand, the integral is then similar to the incomplete Gamma function. https://en.wikipedia.org/wiki/Incomplete_gamma_function

Hope that it helps.


As answered by Karl $$\int t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt=-\frac{t^{\kappa +1} \left(\rho t^{\alpha \kappa +1}\right)^{-\frac{\kappa +1}{\alpha \kappa +1}} \Gamma \left(\frac{\kappa +1}{\alpha \kappa +1},t^{\alpha \kappa +1} \rho \right)}{\alpha \kappa +1}$$ where appears the incomplete gamma function.

This can also write $$\int t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt=-\frac{t^{\kappa +1} E_{\frac{(\alpha -1) \kappa }{\alpha \kappa +1}}\left(t^{\alpha \kappa +1} \rho \right)}{\alpha \kappa +1}$$ where appears the exponential integral function.


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