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The quantity <math display=inline> \lambda(b_i-a_i)</math> can be interpreted as the expected or average number of points occurring in the interval <math display=inline> (a_i,b_i]</math>, namely:
:<math> \operatorname E[N(a_i,b_i
where <math>\operatorname E</math> denotes the [[expected value|expectation]] operator. In other words, the parameter <math display=inline> \lambda</math> of the Poisson process coincides with the ''density'' of points. Furthermore, the homogeneous Poisson point process adheres to its own form of the (strong) law of large numbers.<ref name="Kingman1992page42">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=42}}</ref> More specifically, with probability one:
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