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!



Non-homogeneous Poisson process with intensity function $\lambda\cdot f_{X_1}$
$\newcommand{\la}{\lambda}$
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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Poisson_point_process#Inhomogeneous_Poisson_point_process) with intensity function $\la f\_{X\_1}$.


