# Manipulating a product term inside an integral

I have an expression of the form $$P(x) = \int_0^\infty \prod_{i=0}^{n} e^{-G(a, x_i)}\,\mathrm{d}a$$ and I was wondering if there was any way that I could swap the order of the product and the integral? I suspect not but its been a while since I had to manipulate integrals for myself. Ideally P(x) will represent the total probability of making i measurements from a Possion distributed system with zero occurences each time.

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In general, $\int fg \not= \int f \int g$. –  AlexE Jan 16 '12 at 10:51
Thanks, I thought that was probably the case. –  Bowler Jan 16 '12 at 11:43
1. I think you mean $\prod_{i=0}^N$?! 2. Would it help to write $e^{-\sum G }$? –  draks ... Jan 16 '12 at 13:29

This is a CW answer intended to remove this question from the Unanswered queue.

As AlexE remarks, there is no general way to swap products and integrals.

This means that in general, evaluating integrals like the one stated can be quite hard, especially if the mapping $G$ is not in some way "nice".

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