We have two properties regarding Poisson:
- We know that if we split a Poisson Process (rate=$\lambda$) with probability $p$, we get two sub Poisson Processes that are independent with rates $\lambda p$ and $\lambda(1-p)$ respectively.
- If we have a binomial distribution with probability $p$, and the number of trials is a Poisson random variable (rate=$\lambda$), then this $X=Bin(Poisson(\lambda),p)$ is $Poisson(\lambda p)$.
Suppose I have a Poisson Process (rate=$\lambda$), and the arrival in a unit time $[0,1]$ is therefore a $Poisson(\lambda)$ random variable, let me denote it here as $X_0$.
I would like to define $X_{11}=Bin(X_0,p_1)$ and $X_{12}=Bin(X_0,1-p_1)$. So by property 2, $X_{11}=Poisson(\lambda p_1)$, $X_{12}=Poisson(\lambda (1-p_1))$. My first question is, are $X_{11}$ and $X_{12}$ independent? My guess is that, according to property 1 (Poisson split), they should be independent right?
Now, I further split the Poisson random variables. Define $X_{21}=Bin(X_{11},p_2)$, $X_{22}=Bin(X_{11},1-p_2)+Bin(X_{12},p_2)$, $X_{23}=Bin(X_{12},1-p_2)$. My second question is that are $X_{21},X_{22},X_{23}$ independent?
Thank you so much!