1. ## ask for help on the integral

Anyone can help to calculate the integration of the following function with t? thanks very much~
t is the only variable and others are constant
eq1.JPG

2. Originally Posted by Romsek
what's the question?
to calculate the integration with variable t. sorry for the mistake.

3. anyone can help?..

4. Originally Posted by Romsek
I'm not seeing any sort of wondrous cancellations or anything that would lead this to be able to be integrated to a closed form. Each term of the product has an exponential of a polynomial in t.

$\displaystyle{\prod_{j=1}^n}\left(1-\exp\left[{\displaystyle{-\frac{1}{\lambda^k}\sum_{m=0}^{k-1}}t^m t_j^{k-m}}\right]\right)$

If you expand the whole thing out it will be a big sum of exponentials of polynomials in t which may be able to be evaluated using the product rule and the generalized error function.

I'm guessing this isn't a homework problem. Is this a real world application? Can you just integrate it numerically?
thanks very much for your reply. It is a real application and the integration area is (0,+infinite), is there any approach to integrate it numerically?

5. Originally Posted by Romsek
what I would do is set up the integral in whatever software you plan to do it in and run it using an increasing upper limit of integration. If the integral has any hope of converging you should start to see evidence of that the results of each run start to get closer and closer. Just keep expanding in upper limit until the absolute value of the difference between runs is less than some number you're happy with. A little experience will quickly tell you how far you have to integrate to.

It's a sum of decaying exponentials so it should converge, possible very quickly depending on $\lambda$

If you don't see evidence of convergence well then you have to take a much closer look at things. Cross that bridge when you come to it.
thanks. i will try it. I am also considering whether some of the items in the equation can be simplified and then the equation can be integrated. any comments on this? e.g., to use Taylor expansion?

6. Originally Posted by Romsek
I'm looking at this now and I was wrong. It's not a sum of decreasing exponentials.

I'm using linearly spaced $t_i$'s and playing with k and $\lambda$.

Each term in the product tends towards 1 eventually. You end up with the infinite integral of a constant term. That's not going to converge.

I'll keep playing with it and let you know if I find anything of use.
Look forward to your good news and great thanks for this. Best wishes for the new year~

7. Originally Posted by Romsek
Well thank you.

But I don't have good news. It's pretty clear this thing won't converge.

Can I ask what the actual problem is?
maybe i should put the whole problem here. Give me a moment.

8. Originally Posted by Romsek
Well thank you.

But I don't have good news. It's pretty clear this thing won't converge.

Can I ask what the actual problem is?
The problem description is attached, thanks~
problem def.jpg

9. Originally Posted by Romsek
I'm not going to be able to help you out. I'd hit the technical journal index and search for Weibull order statistics. There's some stuff on google but you'll probably need to dig deeper. There's going to have to be something that keeps t from being allowed to go to infinity.

Someone must have calculated the expectation of the max of a set of Weibull distributed rv's before.
yes, the expectation of the max of a set of weibul distributed variables are already known. However, in my case, the variable Y are not Weibul distribution variables already.. I will try more to solve it. thanks anyway~~

10. Originally Posted by Romsek
another place you might look in in texts on survival analysis.

I'm guessing this problem is related to the remaining lifetime left to a group of components (or people) given that they've survived to various times already.

Good luck!
ok, thanks~

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