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:
$F(i,\tau)=\int_{0}^{\tau-x_{i-1}}e^{-\lambda x_{i}}F(i-1,\tau)$
$F(3,\tau)=\int_{0}^{\tau}\int_{0}^{\tau-x_1}\int_{0}^{\tau-x_2}e^{-\lambda x_1}e^{-\lambda x_2}e^{-\lambda x_3}\ dx_3\ dx_2\ dx_1$
$x_i\ge 0; 1\le i\le n$
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.