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X1, X2 and X3 are three incremental poison processes for time intervals [0,t1], [t1, t2] and [t2,t3] respectively with same rate parameter $ \lambda$

What is the Joint distribution of X1, X2 and X3 given number of success in interval [0, t3] is n ?

I know that the sum of independent X1+X2+X3 ~ Poisson($ 3\lambda$)

So, $ P(X=n)= {e^{-3\lambda} \lambda^{n}}/{n!}$

and $ P(X_i)= {e^{\lambda} \lambda^{-X_i}}/{X_i!}$

so $ P(X_1,X_2,X_3)= {e^{3\lambda} \lambda^{-(X_1+X_2+X_3)}}/{(X_1!X_2!X_3!)}$

I don't have an idea about how to form the conditional PDF

Dom Jo
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  • Could you explain how an "incremental" Poisson process differs from a Poisson process? Are you assuming these processes are independent? (If both of these are the case, collectively you have a single Poisson process of rate $3\lambda$ followed by multinomial selection and the answer can easily be obtained using the solution method for the case of two processes at https://stats.stackexchange.com/questions/429564 .) – whuber Aug 22 '20 at 14:09
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    @whuber yes. but its the jount pdf that I have issue with. – Dom Jo Aug 22 '20 at 14:17
  • @whuber Incremental mean the number of successes for that time interval [ti-1, ti] – Dom Jo Aug 22 '20 at 14:18
  • In a Poisson process with arrival rate $\lambda$, the number of arrivals in an interval of length $T$ is a Poisson random variable with parameter $\lambda T$, and not $\lambda$ as you state. So, $X_1, X_2, X_3$ should be Poisson random variables with parameters $\lambda t_1, \lambda (t_2-t_1), \lambda(t_3-t_2)$ and they are _not_ independent unless you specify the intervals _very carefully_ as $(0,t_1],(t_1,t_2],(t_2,t_3]$ paying particular attention to the placement of $($ and $]$ instead of cavalierly using $[$ and $]$. – Dilip Sarwate Aug 22 '20 at 20:20
  • @Dilip Since the chance of any such event occurring in the set $\{t_1,t_2\}$ is zero, why does the "very careful" specification of the intervals matter? – whuber Aug 22 '20 at 21:10
  • @whuber Assuming as I do that there is only one process and we are looking at three time intervals, then if the time intervals are not _disjoint_ as per the OP's question where they overlap (though only at one point), then it cannot be said that the random variables$X_1, X_2, X_3$ are _independent_ in every instance, They may, for example, share an arrival that occurs at $t_1$ or at $t_2$, In particular, it might be that $X_1=1, X_2=2,X_3=1$ but there are only two arrivals (at exactly $t_1$ and $t_2$) in $[0,t_3]$, that is, $X_1+X_2+X_3$ is _not_ the total number of arrivals in $[0,t_3]$. – Dilip Sarwate Aug 22 '20 at 21:33
  • @Dilip Nevertheless, the random variables *are* independent. An event of measure zero can always be neglected. (Despite the sloppy wording, these variables $X_i$ are *counts* of events in the intervals, not the processes themselves.) – whuber Aug 22 '20 at 21:51
  • @whuber Can it be thought of as processes which start at time t(i-1) and end in time t(i). I noted your point on $\lambda$ being proportional to time. But apart from that, can you help me with the pdf? – Dom Jo Aug 23 '20 at 06:21
  • As explained in the link I originally gave, $(X_1,X_2,X_3)$ has a multinomial distribution. The relative probabilities must be $(t_1\lambda, (t_2-t_1)\lambda, (t_3-t_2)\lambda).$ – whuber Aug 23 '20 at 17:40

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