The Poisson Distribution (AS)

The Poisson Distribution is a discrete probability distribution representing the probability of a given number of events happening in a fixed interval of time.
The Poisson Distribution describes events which are independent of each other, occurring at a uniform average rate.
The events being studied must be exactly defined such as calls per hour, emails per day, etc.
The distribution’s mean (λ) is the average rate of occurrence over the interval being studied.
The probability mass function for a Poisson distribution is given by the formula: P(X = k) = λ^k e^−λ / k!, where P(X = k) is the probability of ‘k’ occurrences in the interval, ‘λ’ is the mean rate of occurrences per interval, ‘k’ is the number of occurrences and ‘e’ is the base of natural logarithms (approx. 2.71828).
In the equation above, ‘k’ must be a non-negative integer and ‘λ’ must be a positive real number.
The mean and variance of a Poisson distribution are both equal to ‘λ’.
A Poisson distribution shows the likelihood for the number of times an event (the ‘k’ term) will happen in an interval, given the ‘λ’ (mean).
The distribution approaches a normal distribution as ‘λ’ becomes large.
Common applications of the Poisson Distribution occur in fields where the times between random events are important, such as telecommunications, astronomy, and finance.