Non-homogeneous Poisson process with intensity function $\lambda\cdot f_{X_1}$ I want to show that there is a non-homogeneous Poisson process with a certain intensity function, but I have some problems while showing that this Poisson process satisfies the axioms(?). I am using the axioms as follows: 


*

*$N(0) = 0$

*if $s\leq t$, then $N(s)\leq N(t)$

*etc


as usual. Now the problem is given by: 
Let $X_1,X_2,\dots$ be i.i.d. continuous random variables with common probability density function $f_{X_1}$ and let $Z \xrightarrow{d}Po(\lambda)$ be independent of $X_1,X_2,\dots$. Now define the random point set $\mathcal{P} = \{X_1,\dots,X_Z\}$ ($\mathcal{P}$ = $\emptyset$ if $Z = 0$). 
Show that $\mathcal{P}$ is an inhomogeneous Poisson point process with intensity function $\lambda\cdot f_{X_1}$.
Sorry if my question is not clear enough, and thank you!
 A: $\newcommand{\la}{\lambda}$
Take any $t_0,\dots,t_n$ such that $0=t_0<\dots<t_n=\infty$. For each $j\in[n]:=\{1,\dots,n\}$, let $I_j:=[t_{j-1},t_j)$ and 
\begin{equation}
 \nu_j:=\#\{i\in[Z]\colon X_i\in I_j\}. 
\end{equation}
Then, for each nonnegative integer $z$, the joint conditional distribution of $(\nu_1,\dots,\nu_n)$ given $Z=z$ is the multinomial distribution with parameters $z,p_1,\dots,p_n$, where 
\begin{equation}
 p_j:=P(X_1\in I_j). 
\end{equation}
So, for any nonnegative integers $k_1,\dots,k_n$
\begin{align}
 P(\nu_1=k_1,\dots,\nu_n=k_n)
 &=\sum_{z=0}^\infty P(Z=z)P(\nu_1=k_1,\dots,\nu_n=k_n|Z=z) \\ 
 &=\sum_{z=0}^\infty \frac{\la^z e^{-\la}}{z!}
 \frac{z!}{k_1!\cdots k_n!}\, p_1^{k_1}\cdots p_n^{k_n}\,1\{k_1+\dots+k_n=z\} \\ 
 &=\prod_{j\in[n]}
 \frac{(\la p_j)^{k_j}e^{-\la p_j}}{k_j!}. 
\end{align}
So, $\nu_1,\dots,\nu_n$ are independent random variables and $\nu_j\sim Poisson(\la p_j)$ for each $j\in[n]$. 
This means that indeed $\mathcal{P}$ is an inhomogeneous Poisson point process with intensity function $\la f_{X_1}$.
A: Basically a Poisson point process can be defined on the real line by considering the number of points of the process in the interval (a.b], which is \lambda*(F(b)-F(a)) (\X_i s are iid), then use the definition of the intensity… 
