Integrating the product of Poisson and exponential pdf So I'll spare the background as to why, but I'm trying to integrate the following:
$$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$
If you parameterize a Poisson w/ $\lambda$ and an exponential w/ $\mu$ and multiply their pdf's, you get the above.  I just can't seem to do the integration by parts.
Is this a two step IBP?  Is there an easier way to solve than actually integrating (product of random variables, etc.)?

I'm not sure if I understand $\mu$ and $\lambda$, but if they're just constants we have a $n$ step integration by parts in $t$:
$$ \frac{\lambda^n}{n!} \int_0^\infty e^{-(\lambda + \mu) t }  t^n dt =\frac{\lambda^n}{n!} \left ( -\frac{ e^{- ( \lambda + \mu ) t } t^n }{\lambda + \mu } \Big |_0^\infty +
 \frac{n}{\lambda + \mu } \int_0^\infty  e^{- ( \lambda + \mu ) t } t^{n-1} dt \right ) =  \frac{n \lambda^n }{(\lambda + \mu) n! } \int_0^\infty  dte^{- ( \lambda + \mu ) t } t^{n-1}   $$ 
Just rinse and repeat.
