Discrete Random Variables: The Poisson distribution
Discrete Random Variables: The Poisson distribution
Poisson Distribution Overview
- The 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 constant mean rate and independently of the time since the last event.
- The distribution is named after French mathematician Siméon Denis Poisson.
Defining Characteristics
- The mean and variance of a Poisson distribution are both equal to λ, the rate of occurrence of the event.
- The number of outcomes is potentially infinite, i.e., it can be any non-negative integer: 0, 1, 2, 3, and so on.
Probability Mass Function
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The probability mass function of a Poisson distribution is given by:
P(X=k) = λ^k * e^(-λ) / k!
where:
- P(X=k) is the probability that there are k occurrences,
- λ is the expected number of occurrences,
- e is the base of the natural logarithm (approximately equal to 2.71828),
- k! is the factorial of k.
Common Assumptions
- The Poisson distribution often involves making several assumptions including:
- Events are independent of each other.
- The average rate (λ) is constant across the observed timeline.
- Two events cannot occur at exactly the same instant; in other words, the probability of more than one occurrence in an infinitesimally small interval is negligible.
Applications
- The Poisson distribution is used in many areas such as telecommunication (for packet traffic), finance (for number of trades), astronomy (for number of stars in a galaxy), and manufacturing (for defect count). It is also used in general statistical studies where the event frequency is of interest.
Limitations
- If the assumptions of independence and constant rate (λ) are violated, a Poisson distribution may not be the appropriate model.
- For large sample sizes or large mean rates (λ), the data may be better described by a different distribution, such as the normal distribution.