Why do we require conditions for the Poisson Distribution?

[chi_rho]

Three conditions must be met in order for the Poisson Distribution to be used:

1) The average count rate is constant over time
2) The counts occurring are independent
3) The probability of 2 or more counts occurring in the interval $n$ is zero

Simply, why must these conditions be met for valid use of the Poisson Distribution?

 

[mathman]

The first two are based on the definition. I am not sure what the third condition is, but being independent the counts may be arbitrarily close..

 

[Heinera]

3) seems to be something related to some discrete approximation to the Poisson distribution, not pertaing to the proper distribution per se.

The other two are simply properties of the distribution. If they don't hold, the data aren't Poisson distributed in the first place.