# Split Poisson Process AND severity

I have a Poisson process whose statistics are interarrival times ($\bf X$), number of arrivals ($\bf N$), and arrival times ($\bf T$). Later, the process is split by a Bernoulli process that categorizes each event with a probability $p$ of hit, and $1-p$ of miss.

As a result I get two separate Poisson processes, one for hit ($\textbf{X}_h$,$\textbf{N}_h$,$\textbf{T}_h$) and the other for miss ($\textbf{X}_m$,$\textbf{N}_m$,$\textbf{T}_m$). But I'm not just happy with having both process categorized, I want to assign a probabilistic "severity" degree to hit events.

So suppose, (taking the example from this website) that the Poisson arrivals are radioactive emissions and each emitted particle is either (by Bernoulli) detected (type 1 with probability $p$) or missed (type 0 with probability $1-p$) by a counter:

My question: Is it possible to assign a quantity, like velocity, to the detected particle in the example, by sampling a proper density function?

UPDATE

Maybe what I'm asking is: given that a Poisson process ($\lambda$) is preservable under random selection when counted with probability $p$, and that the thinned process is still Poisson with parameter $\mu=\lambda p$, maybe $p$ can involve the probabilities of two events hapenning, sthg like $p = p_1 \times p_2$, i.e. partitioning $p$ under some criteria. Thus $\mu = \lambda(p_1p_2)$ Does it make sense?

UPDATE 2

Provided that the "thinning probability" $p$ must satify the binomial probability law $\binom {n} {k} p^k q^{n-k}$. Now, say that I take the example from Parzen's, Stoch Proc, 48: "Suppose that customers pass by a shop in accord with a Poisson'process at mean rate $\lambda$. If each customer has probability $p$ of entering the shop, customers enter the shop in accord with a Poisson process with mean rate $\mu=\lambda p$". Now, if I also know that entering customers will also buy cookies with prob $p_1$, how should I factor in this new piece of information to describe the new stochastic process? (PS. It doesn't matter if the new thinned process with isn't Poisson anymore.)"

Thanks!

• (1) Could you elaborate on what is meant by "probabilistic severity degree"? (2) Usually quantities like $X$, $N$, and $T$ do not "define" a Poisson process--they are statistics that can be derived by observations of it and are closely inter-related. Do you perhaps have in mind some kind of generalization of Poisson processes?
– whuber
Dec 1, 2014 at 22:20
• Hi whuber, I updated the post. please check and thanks for the clarification! Dec 1, 2014 at 22:34
• Thanks--that clears up a lot. But now the appearance of "velocity" is surprising. To what property of the process does that refer?
– whuber
Dec 1, 2014 at 22:49
• To no property of the process I guess, but it's related to the physical problem. I'm interested in knowing if there's a framework that contemplates such superposition of different pieces of information... Dec 1, 2014 at 22:52
• The difficulty we're facing here is that up until this comment you haven't provided any information at all to connect the statistical framing of the question (as a thinned Poisson process) to any physical characteristics of the system it is supposed to be describing. It is hard to see how your question could be answered, much less even understood, without such information. Perhaps you could be more specific about what you mean by "superposition" of "pieces of information" and how that might relate to the specific circumstances you contemplate?
– whuber
Dec 1, 2014 at 22:54