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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

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  • $\begingroup$ 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 stats.stackexchange.com/questions/429564 .) $\endgroup$
    – whuber ♦
    Commented Aug 22, 2020 at 14:09
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    $\begingroup$ @whuber yes. but its the jount pdf that I have issue with. $\endgroup$
    – Dom Jo
    Commented Aug 22, 2020 at 14:17
  • $\begingroup$ @whuber Incremental mean the number of successes for that time interval [ti-1, ti] $\endgroup$
    – Dom Jo
    Commented Aug 22, 2020 at 14:18
  • $\begingroup$ 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 $]$. $\endgroup$
    – Dilip Sarwate
    Commented Aug 22, 2020 at 20:20
  • $\begingroup$ @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? $\endgroup$
    – whuber ♦
    Commented Aug 22, 2020 at 21:10

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