In some test, I've seen the affirmatives (regards to poisson distribution.):
1 - The probability of an occurrence is the same across the field of observation.
2 - The probability of more than one occurrence at a single point is approximately zero
3 - The number of occurrences in any interval is independent of the number of occurrences in other intervals.
So, the first is ok to me, but in 2 and 3 I'm in doubt.
2 is because $n = 1$, right? and $k \ge 2$, so $$P(k) = \frac{e^{-\lambda}.\lambda^{k}}{k!}$$ where $\lambda < 1$.
And in (3) is where I don't understand how a occurance is independent once, for instance, if we have 5 calls in 10 min, in 20 min we have 10 calls.. how is it independent? I think It can be a misundastanding of me.
Can someone elucidate?
This looks to me like a Poisson process rather than a Poisson distribution. It has various properties:
The gap between successive events has an exponential distribution with some rate parameter $t$, i.e. if $X$ is the gap then $\mathbb P(X\le t)=1-e^{-\lambda t}$ for $t\ge 0$
So the probability that a gap is $0$ or less (and it cannot be negative) is $1-e^0=1-1=0$ meaning the probability of more than one occurrence at a single point where it occurs at least once is zero, which is point (2)
The number of events in a interval of length $t$ has a Poisson distribution with mean $\lambda t$
Since the exponential distribution is memoryless, the numbers of events in two non-overlapping intervals are independent, i.e. point (3)
Another consequence of the memoryless property is that all the points in a given interval have the same likelihood of occurring, which is point (1)