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I have a Poisson process $N(t)$ with $\tau$ for customer arrival in a shop. $N(t)$ is spllitted with two types of arrival (male and female). It can be shown that the process is a combination of two types as Poisson processes $M(t)$ and $F(t)$ with $p \lambda$ and $(1-p) \lambda$.

It can proved conditioning on the number of the total arrivals and using the concept of binomial distribution (n,p). Now I have the proof of that already.

But I want to see prove that both $M(t)$ and $F(t)$ satisfies the properties of the Poisson process, i.e :

Given,

  1. $P(N(t) = 1 )= \lambda h+ o(h)$

  2. $P (N(t) > 1) = o(h)$

How can I find $P(M(t) = 1 )$ and $P(M(t) > 1 )$ ?

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1 Answer 1

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Use your words.

1) If no arrivals of any type have occurred at time zero, how many of either type have occurred?

2) Having stationary, independent increments, means the distribution of arrivals in a period is dependent only on the length of the period, not on its time of start (nor on amount of arrivals before the period).   Since the distribution of amount of each type of arrival within a period is dependent only on total amount of arrivals (within that period), then...

3) We know $P(N(t)=1)= \lambda h + o(h)$, and $P(N_1(t)=1\mid N(t)=1)=p$, so...

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  • $\begingroup$ thanks for the answer. I stil have some confusion actually. How to find, $P(N_1(t) = 1 | N(t) > 1) $ =? and $P(N_1 (t) \geq 2 )$? $\endgroup$
    – jhon_wick
    Commented Aug 4, 2015 at 1:28

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