# How to calculate $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$?

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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• Are the limits of the integral $(-\infty,\infty)$? If so, and $k=1$, then I sense an application of the Cauchy residue theorem. – Peter Leopold Mar 1 at 21:59
• @PeterLeopold, No it's a definite integral. – gultu Mar 1 at 22:00
• 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. – 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 )
}

I(0,1,0,1,1,0,1,1,1,2)


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

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