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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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