We are given that $X1,X2$ are i.i.d. continuous random variables with common pdf $f_{X1}$. Let $Z \stackrel{d}{=} Po(λ)$ be independent of $X1,X2,\ldots$ . Now define the random point set $\mathscr{P} = \{X1,\ldots,X_Z\}$
(So $\mathscr{P} = \emptyset$ if $Z = 0$).
Show that $\mathscr{P}$ is an inhomogeneous Poisson process.
What is the intensity function of the inhomogeneous Poisson process $\mathscr{P}$ ?
$\textbf{Definition:}$ An intensity function for the inhomogeneuous Poisson process $\mathscr{P}$ is a function
$f:\mathbb{R}\to[0,\infty)$ such that: $$N(A)\stackrel{d}{=} Po\left( \int_{A} f(t)dt\right),$$
where $N(A) := \#\{\text{arrivals in }A\}$.
I am quite certain that the intensity function must be $λ \cdot f_{X1}$, but I am not sure how to prove it. Can somebody please help me with this? Thanks in advance.
I preassume that it is taken for granted already that $N(A)$ has indeed Poisson distribution so that it is enough to find its parameter (which equals its expectation).
Observe that: $$N\left(A\right)=\sum_{k=1}^{Z}\mathsf{1}_{A}\left(X_{k}\right)$$ so that for nonnegative integer $n$:
$$\begin{aligned}\mathbb{E}\left[N\left(A\right)\mid Z=n\right] & =\mathbb{E}\left[\sum_{k=1}^{Z}\mathsf{1}_{A}\left(X_{k}\right)\mid Z=n\right]\\ & =\mathbb{E}\left[\sum_{k=1}^{n}\mathsf{1}_{A}\left(X_{k}\right)\mid Z=n\right]\\ & =\mathbb{E}\sum_{k=1}^{n}\mathsf{1}_{A}\left(X_{k}\right)\\ & =nP\left(X_{1}\in A\right)\\ & =n\int_{A}f_{X_1}\left(t\right)dt \end{aligned} $$
This tells us that: $$\mathbb{E}\left[N\left(A\right)\mid Z\right]=Z\int_{A}f_{X_1}\left(t\right)dt$$ A consequence of this is: $$\mathbb{E}N\left(A\right)=\mathbb{E}\left[\mathbb{E}\left[N\left(A\right)\mid Z\right]\right]=\mathbb{E}Z\int_{A}f_{X_1}\left(t\right)dt=\lambda\int_{A}f_{X_1}\left(t\right)dt=\int_{A}\lambda f_{X_1}\left(t\right)dt$$This proves that $\lambda f_{X_1}$ can be marked as intensity function.