The Poisson distribution is the discrete distribution that is one of the most important limiting cases of the binomial distribution. With the growth n and fixed value of the product np=λ > 0 the binomial distribution B(n,p) converges to the Poisson distribution.
The function of probability density is given by the formula:
Where:
λ. Rate of events is a positive real number, equal to the expected number of occurrences during the given interval.
k. The number of occurrences of an event, the probability of which is given by the specified function.
To model random values, use an algorithm based on a ratio of the Poisson distribution with the λ parameter and exponential distribution with the parameter (1/λ). Thus, the following operations are executed to draw a sample from pseudo-random value X ~ P(λ):
1. Assume that a = e–λ, b = 1 and i = 0.
2. Generate a pseudo-random variable ui+1 ~ U(0, 1) from the continuous uniform distribution at the interval (0, 1), and after this assign the values b = b·ui+1.
3. If b < a, the value x = i is returned, otherwise the algorithm returns to the step 2 with i = i + 1.
See also:
ISmPoissonDistribution | IStatistics.Poisson | Library of Methods and Models