Poisson distribution

Introduction to Poisson Distribution

  • The Poisson distribution is a type of probability distribution that measures the probability of a given number of events happening in a fixed space or time.
  • It’s named after the French mathematician Siméon Denis Poisson.
  • This distribution is often used in scientific and mathematical research, in fields like biology, chemistry, physics, and economics.

Characteristics of Poisson Distribution

  • It can only handle integer values (0,1,2,3…), meaning that events must be countable and cannot be fractional.
  • Each event must be independent of the others, meaning that one event occurring doesn’t influence whether another event occurs.
  • The average rate (λ - lambda) of events per interval must remain constant.

Using Poisson Distribution

  • You use the Poisson distribution formula, P(X=x) = λ^x * e^-λ / x!, to calculate the probability of a specific number of events.
  • In this formula, X is the actual number of successes that result from the experiment, e is a constant equal to approximately 2.71828, λ is the mean number of successes that occur in a specified region, and x! is the factorial of x.
  • It’s common to use Poisson distribution to model phenomena such as the number of telephone calls at a call centre in a given time period, the number of decay events in radioactivity per given time unit, or the number of emails received by an email server in a day.

Expectation and Variance of Poisson Distribution

  • The expectation (mean) of a Poisson distribution is λ.
  • The variance of a Poisson distribution is also λ.
  • The standard deviation is the square root of λ.

When to Apply Poisson Distribution

  • Poisson distribution is used when the events are random, independent, and occur at a constant average rate within a specific spatial or temporal context.
  • It’s typically used with a large sample size and a small probability of success.
  • Note, however, that while Poisson distribution is useful, it’s a simplification and may not always provide an accurate model of reality. Often, you’ll need to consider other factors or more complex models, depending on the specific situation.

Further Points

  • A thorough understanding of Poisson distribution is crucial for the Further Stats 1 part of the maths Further Maths syllabus.
  • It sets a fundamental basis for comprehending more advanced statistical topics such as exponential distribution and queueing theory.