poisson process

**shilz222** · January 30th 2008, 03:15 AM

Two of the criteria for a counting process to be Poisson are the following: (a) $P \{N(h)= 1 \} = \lambda h + o(h)$ and (b) $P \{N(h) \geq 2 \} = o(h)$ .

So this is saying that the probability that the count exceeds $2$ is essentially $0$ ?

**a tutor** · January 30th 2008, 04:29 AM

Originally Posted by shilz222

Two of the criteria for a counting process to be Poisson are the following: (a) $P \{N(h)= 1 \} = \lambda h + o(h)$ and (b) $P \{N(h) \geq 2 \} = o(h)$ .

So this is saying that the probability that the count exceeds $2$ is essentially $0$ ?

Exceeds 1 in fact...

Isn't it just saying that you don't have simultaneous events so as you look at smaller and smaller time intervals the probability of more than 1 event -->0

It's a long time since I did this myself so beware.

**shilz222** · January 30th 2008, 08:44 AM

I see. In small intervals of time, an the probability of more than one event occurring decreases (this is from an intuitive standpoint).

But for example, can this be applied to real world counting processes.

And $o(h)$ is a function $f(\cdot)$ defined by $\lim_{h \to 0} \frac{f(h)}{h} = 0$ .

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