The general form of a Poisson process (or a Levy process) can be defined as; the number of events in time interval $(t, t + T]$ follows a Poisson distribution with associated parameter λT. This relation is given as
\begin{equation}
P[(N(t+T)-N(t))=k] = \frac{(\lambda T)^k e^{- \lambda T}}{k!}
\end{equation}
Where $N(t + T) − N(t) = k$ is the number of events in time interval $(t, t + T]$. It will be obvious to you that $\lambda$ is the rate parameter. Assume in the simplest case for some $T$ where $k=1$ then
\begin{equation}
f(\lambda) = \lambda e^{-\lambda T}
\end{equation}
Then taking Laplace transforms yields
\begin{equation}
\widehat{f(t)} = \frac{\lambda}{\lambda + s}
\end{equation}
I'll leave it to you to fill in the more specific details, I've only dropped the textbook conclusions from a Poisson process and the Laplace transform of such a process.