Poisson Distribution

Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ? 100 and np ?10.
-If for every t > 0 the number of arrivals in the time interval [0,t] follows the Poisson distribution with mean ? t, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1 / ?
-The cumulative distribution functions of the Poisson and chi-squared distributions are related in the following ways:

and