A Poisson distribution is appropriate when you have events that occur
The last condition means that the probability of more than one event occurring at any particular point in space or time is zero (or negligible).
However, if the parameter (mean) of the Poisson distribution, L, is greater than 10, you may be better off using the Normal approximation, N(L, L), with the appropriate continuity corrections.
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Why belong exponential family for poisson distribution
A poisson process is a non-deterministic process where events occur continuously and independently of each other. An example of a poisson process is the radioactive decay of radionuclides. A poisson distribution is a discrete probability distribution that represents the probability of events (having a poisson process) occurring in a certain period of time.
The exponential distribution and the Poisson distribution.
Given a Poisson distribution with mean = 2. Find P(X < 5)