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Is there any simplified closed form for the following integral?

$\int_0^{\tau}\int_0^{\tau-x_1}\int_0^{\tau-x_2}\dots\int_0^{\tau-x_{n-1}}e^{-\lambda x_1}e^{-\lambda x_2}\dots e^{-\lambda x_n}\ d{x_n}\ \dots d{x_2}\ d{x_1}$

in which

$x_i\ge 0; 1\le i\le n$

We can also define it recursively as follows if it helps:

$F(i,\tau)=\int_{0}^{\tau-x_{i-1}}e^{-\lambda x_{i}}F(i-1,\tau)$

$F(3,\tau)=\int_{0}^{\tau}\int_{0}^{\tau-x_1}\int_{0}^{\tau-x_2}e^{-\lambda x_1}e^{-\lambda x_2}e^{-\lambda x_3}\ dx_3\ dx_2\ dx_1$

$x_i\ge 0; 1\le i\le n$

Alireza Montazeri Gh
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    I don't think it makes sense to have your dummy variable the same as the variable in your limits of integration. I'm happy to stand corrected though. – Matthew Cassell Jun 07 '16 at 04:23
  • I just fixed it. Does it make sense now? @Mattos – Alireza Montazeri Gh Jun 07 '16 at 04:29
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    "Does it make sense now?" Not really: the integration bounds for $x_2$, say, are really $0$ and $x_1-\tau<0$, or do you mean $\tau-x_1$? Likewise for the other $x_k$-bounds. – Did Jun 07 '16 at 07:01
  • OMG, I meant $\tau -x_i$ @Did – Alireza Montazeri Gh Jun 07 '16 at 07:07
  • And now, are you sure that $\tau-x_2$ should not be $\tau-x_2-x_1$ instead? Likewise for the other upper bounds, until $\tau-x_{n-1}-\cdots-x_2-x_1$ instead of $\tau-x_{n-1}$? – Did Jun 07 '16 at 07:09
  • I'm pretty sure @Did What I am trying to solve is as follows: – Alireza Montazeri Gh Jun 07 '16 at 07:13
  • Are you after $P(\max{X_1+X_2,X_2+X_3,\ldots,X_{n-1}+X_n}<\tau)$, with $(X_k)$ i.i.d. exponential with parameter $\lambda$, by any chance? – Did Jun 07 '16 at 07:15
  • We have a system in which events happening one after each other. We show each event by $X_i$. Then, $X_1, X_3 \dots X_n$ as i.d.d random variables all having exponential distribution $E(X_i)=\frac{1}{\lambda}$. Having $n$ events in the system, I need to find the PDF of average time for the system when the average time betweren arrival of event $X_{i-1}$ and arrival of event $X_{i+1}$ ($1\le i \le n-1$)is less than $ \tau$. @Did – Alireza Montazeri Gh Jun 07 '16 at 07:28
  • Well, the least one can say is that this is not what your question asks... – Did Jun 07 '16 at 08:56
  • Could you please take a loot at my question in the following link: @Did http://math.stackexchange.com/questions/1817387/finding-pdf-for-a-system-with-n-events-all-having-exponential-distribution – Alireza Montazeri Gh Jun 07 '16 at 17:07

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Since this is a continuous function on a smooth boundary in $R^n$, you can simply use Fubini's theorum to express the n-dimensional multiple integral into a product of n-single integrals on $R$.

$\int_0^{\tau}\int_0^{x_1-\tau}\int_0^{x_2-\tau}\dots\int_0^{x_{n-1}-\tau}e^{-\lambda x_1}e^{-\lambda x_2}\dots e^{-\lambda x_n}\ d{x_n}\ \dots d{x_2}\ d{x_1}$

= $\int_0^{\tau}e^{-\lambda x_1}\ d{x_1}$ *$\int_0^{x_1-\tau}e^{-\lambda x_2}d{x_2}$ *$ \dots$ $\int_0^{x_{n-1}-\tau}e^{-\lambda x_n}\ d{x_n}$

We simply then integrate each expression beginning with the one on the far left.

Ick.

Very tedious, but not difficult.

Mathemagician1234
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