The Poisson distribution is a discrete probability distribution that describes the number of occurrences of an event in a fixed interval of time or space, assum...
The Poisson distribution is a discrete probability distribution that describes the number of occurrences of an event in a fixed interval of time or space, assuming that the events are independent and occur with a known average rate.
It is commonly used in situations involving counting the number of occurrences of a specific event, such as the number of arrivals at a bus stop, the number of defects in a manufactured product, or the number of hits on a dartboard.
The probability mass function of the Poisson distribution is given by:
where:
(X) is the random variable representing the number of events that occur in a given interval.
(\lambda) is the average number of events that occur in that interval.
(k) is the number of events that we are interested in counting.
(e) is the base of the natural logarithm.
(k!) is the factorial of (k).
The mean and variance of the Poisson distribution are given by:
The Poisson distribution is a limiting distribution for the number of occurrences of a random variable with a finite number of possible outcomes, when the number of outcomes is large and the average number of occurrences is finite.
The Poisson distribution is a versatile tool that can be used to model a wide variety of real-world phenomena. It is a powerful tool for understanding and predicting the number of occurrences of an event in a given interval of time or space