1

Denote $S_n$ is the arrival time of the $n$ arrival, and $X_{n+1}$ is the waiting time between the $n$th arrival and the $(n+1)$th arrival in a Poisson process. I want to ask of the independence of $S_n$ and $X_{n+1}$. Why are they independent?

I saw in some books an explanation for it "because the distribution of $S_n$ can be specified by the joint distribution of $X_1,X_2,\ldots, X_{n}$, and those are independent of $X_{N+1}$, then $S_n$ and $X_{n+1}$ are independent". But I do not understand this point clearly. How is the distribution of $S_n$ specifed, specifically?

Moreover, is it true that if random variables $X$ and $Y$ are indepedent, $X$ and $Z$ are independent, then $X$ and $Y+Z$ are independent?

Maybe my questions look quite trivial, I really need specific explanations for them. Thanks in advance.

Tien Kha Pham
  • 3,000

1 Answers1

4

Recall the following elementary lemma:

Let $(X_1,\ldots,X_{n+1})$ be (jointly) independent random variables and $g: \mathbb{R}^n \to \mathbb{R}$ be a measurable function. Then $g(X_1,\ldots,X_n)$ and $X_{n+1}$ are independent.

By definition, we have $S_n = \sum_{i=1}^n X_i$; therefore, setting $$g(x_1,\ldots,x_n) := \sum_{i=1}^n x_i$$ we obtain from the previous lemma that $S_n = g(X_1,\ldots,X_n)$ is independent of $X_{n+1}$.

Concerning your second question: No, this is in general not true since pairwise independence does not imply joint independence. (If $X,Y,Z$ are jointly independent, then $X$ and $Y+Z$ are independent.)

saz
  • 123,507
  • Excuse me, could you give me some articles or books that has the proof for the above lemma? – Tien Kha Pham Jun 10 '15 at 15:25
  • @TienKhaPham Well, the proof is contained in any book on probability theory, I would say. Here you can find a proof for the case $n=2$; if you don't get along with it, I can add a proof to my answer. – saz Jun 10 '15 at 15:33
  • I've just learnt measure theory at a very basic level, so it's really hard for me to understand the proof. Is there any proof that does not need measure theory? – Tien Kha Pham Jun 11 '15 at 12:57
  • @TienKhaPham I don't think so. How can you talk about measures, random variables, ... if you don't know about measure theory? This is a mystery to me. – saz Jun 11 '15 at 17:18
  • I learnt the first course in probability this semester but my prof told me to do an article of Poisson process. I have just learnt it myself in the last month, so it is not enough time for me to understand it deeply. That's the reason why some of the (trivial) results are quite unfamiliar with me (my question is an example). Therefore I tried to find non-measure theory proof for all of the results. – Tien Kha Pham Jun 12 '15 at 02:40
  • @TienKhaPham I see. I still don't see how to prove it without measure theory; sorry. Good luck. – saz Jun 12 '15 at 04:38