A Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. Its characteristics include:

1. Discrete Data: It deals with discrete data, meaning the data can only take certain values (typically non-negative integers).
2. Independence: Events are independent; the occurrence of one event does not affect the probability of another event occurring.
3. Constant Mean Rate: The average rate at which events occur is constant over the given interval. This rate is denoted by λ (lambda).
4. Randomness: Events occur randomly and uniformly throughout the interval.
5. Non-Negative Integer Values: The distribution gives the probability of observing 0, 1, 2, 3, ... events.
6. Well-Defined Probability Mass Function: The probability of observing exactly k events is given by the formula:

   P(X = k) = (λ^k * e^(-λ)) / k!

   where:

   • λ is the average rate of events.
   • e is Euler's number (approximately 2.71828).
   • k is the number of events.
   • k! is the factorial of k.
7. Mean and Variance: The mean (average) and the variance of a Poisson distribution are both equal to λ.
8. Applications: It's often used to model rare events such as:

   • The number of phone calls received by a call center per hour.
   • The number of cars passing a point on a highway per minute.
   • The number of defects in a manufactured product.