Poisson Distributions

Understanding Poisson Distributions

  • A Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space.
  • It’s used when events occur at a constant mean rate and independently of the time since the last event.
  • The Poisson process assumes that the number of events is proportional to the length of time, there is no simultaneous occurrence of events, and previous events do not influence future events.
  • The probability mass function is given by P(X = k) = λ^k * e^-λ / k! where ‘λ’ is the mean number of events per interval and ‘k’ refers to the actual number of successes that result from the experiment.

Calculating Probabilities with Poisson Distributions

  • Making use of the probability mass function, you can calculate the likelihood of ‘k’ events occurring in an interval.
  • Be cautious with the value of ‘λ’. If an event is expected to be observed, on average, ‘r’ times in an interval of length ‘t’, then ‘λ’ is typically ‘r*t’.
  • Remember to utilise factorial function in your calculations when it comes to ‘k!’, and ‘e’ is approximately equal to 2.71828, the base of natural logarithm.

Properties and Applications of Poisson Distributions

  • The expected value of a Poisson distribution (E[X]) is λ. The variance of a Poisson distribution (Var[X]) is also λ.
  • Poisson distribution principles can be visible across several areas, including modelling the number of phone calls received by a call centre per hour, or the number of decays in a given amount of radioactive material in a fixed interval.
  • A key feature of Poisson distributions is that as ‘λ’ grows larger, the distribution starts to resemble a normal distribution. This is referred to as the Poisson approximation to the normal.

Key Points to Remember

  • Take note that events in Poisson distribution are independent.
  • In scenarios where the rate of occurrences is of interest and events occur within a constant time or space, Poisson distributions are incredibly useful.
  • The mean and variance being equal is a distinctive characteristic that can help to identify when to use Poisson distribution in problem-solving.