How can I calculate the integral- $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$ where $f(t)= \mu_0+\mu_1 e^{-\gamma (7+logt)^\delta}$ and $g(t)= \frac{1}{(t+h)^k}$ .

Here $\Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)= \frac{1}{\sigma \sqrt{2 \pi} } \int_{-\infty}^{l_0} e^{\frac{(m-f(t))^2}{2 \sigma^2}} dm$; and $l_0,\mu_0,\mu_1,\gamma,\delta,h,k$ are constants.


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  • $\begingroup$ Are the limits of the integral $(-\infty,\infty)$? If so, and $k=1$, then I sense an application of the Cauchy residue theorem. $\endgroup$ – Peter Leopold Mar 1 at 21:59
  • $\begingroup$ @PeterLeopold, No it's a definite integral. $\endgroup$ – gultu Mar 1 at 22:00
  • 1
    $\begingroup$ Welcome to the site. Could you explain how this integral arises? The question has been flagged as off-topic, presumably since the connection to statistics is unclear. $\endgroup$ – Juho Kokkala Mar 2 at 11:57

Use the integrate() command in R.

I = function(mu0, mu1, L0, gamma, d, h, k, sigma, S, T) 
  f = function(t) mu0 + mu1*exp(-gamma*(7 + log(t))^d) 
  g = function(t) 1/( (t+h)^k ) 
  z = function(t) dnorm( (L0-f(t))/sigma ) * g(t) 
  return( integrate(z, S, T)$val ) 


Make sure the arguments result in defined function values for the integrand (e.g. S$\geq$0).

  • $\begingroup$ Thanks but I need analytical result. Not R command. $\endgroup$ – gultu Mar 1 at 21:55
  • $\begingroup$ Good luck with that @gultu. Again, what is the statistical question? $\endgroup$ – beta1_equals_beta2 Mar 1 at 21:57
  • $\begingroup$ @gultu If you need an analytical result, why did you include the numerical-integration tag on your post? $\endgroup$ – Sycorax Mar 1 at 22:08
  • $\begingroup$ @Sycorax, sorry just noticed. Deleting right now $\endgroup$ – gultu Mar 1 at 22:09

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