The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space R d, for which D , the area, volume or, more generally, the Lebesgue measure of the region is finite, and if N ( D ) denotes the number of points in D, then.
![Large Large](/uploads/1/2/5/6/125614317/794658971.jpg)
.In, the law of large numbers ( LLN) is a that describes the result of performing the same experiment a large number of times. According to the law, the of the results obtained from a large number of trials should be close to the, and will tend to become closer as more trials are performed.The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game.
![Process Process](/uploads/1/2/5/6/125614317/697473898.png)
It is important to remember that the law only applies (as the name indicates) when a large number of observations is considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be 'balanced' by the others (see the ). 1)(Lebesgue integrability of X j means that the expected value E( X j) exists according to and is finite. It does not mean that the associated probability measure is with respect to.)An assumption of finite Var( X 1) = Var( X 2) =.