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.)?

You can use the Laplace Transform. The transform of $t^n$ is $$\frac{n!}{s^{n+1}}$$ So you get, after some algebra, the quantity must equal $$\lambda^n {n!\over n!}\frac{1}{((\lambda +\mu))^{n+1}}=\frac{\lambda^n}{(\lambda+\mu)^{n+1}}$$ Can someone check this please? And evidently you've made a mistake somewhere because that is clearly not a probability function.
