The Poisson distribution

The Poisson Distribution

Definition and Properties

  • The Poisson distribution is a probability distribution which expresses the probability of a given number of events happening in a fixed interval of time or space.
  • Each attempt for an event to occur is independent of each other; thus, the happening or not happening of a previous event doesn’t affect the probability of the next event.
  • The average number of events occurring is represented by λ (lambda), which is the mean and variance of the distribution.
  • The events are described as being rare on the scale of interest, i.e., the probability of an event occurring is relatively small.
  • The exact timing between events (inter-event time) follows an exponential distribution.

  • The probability mass function for the Poisson distribution is given by:

    P(X=k) = λ^k e^-λ / k! for k = 0, 1, 2, ….

where P(X=k) is the probability of k events occurring in an interval, λ is the rate at which events occur, e is the base of the natural logarithm (approximately equal to 2.71828), k! is the factorial of k.

Utilisation in Mathematical Problems

  • The Poisson distribution is applied to problems requiring the calculation of probabilities for number of events occurring within any given time period.
  • Example fields of use include queuing theory, reliability theory, and telecommunication traffic or signal processing.

Key Assumptions and Limitations

  • It assumes that the events occur independently, i.e., an event doesn’t affect the probability that another event will occur.
  • The fact that the Poisson distribution assumes that the mean and the variance are the same can be limiting since they are not always identical in real-life data.
  • It assumes that events can occur any number of times during an interval of interest; it doesn’t model situations where, after an event has occurred, a certain amount of time must pass before another event can occur.
  • It assumes that the probability of an event happening is the same for any two intervals of the same length. This assumption might not hold true in various situations.

Connection with Other Distributions

  • The Poisson distribution is related to the binomial distribution; when the number of trials in a binomial distribution approaches infinity, and the expected number of successful trials remains fixed, it approximates a Poisson distribution.
  • The inter-event times in a Poisson process are exponentially distributed.

Poisson distribution is a vital part of the distribution theory as it enables us to deal effectively with a range of practical problems, particularly when dealing with events which occur independently and at a constant average rate.