Probability Distribution- Poisson: The Poisson distribution is a discreet probability distribution that expresses the probability of occurrence of a given number of events in occurring within a given fixed interval. For example, the number of defects per batch in a production process or the number of calls per hour arriving at the 911 emergency switchboard are discrete random variables that follow a Poisson distribution.
The mathematical expression for the Poisson distribution for obtaining X successes, given that X successes are expected, is:
While the Poisson random variable K refers to the number of successes per unit, the parameter lambda (X) refers to the average or expected number of successes per unit.
Assumptions for Poisson distribution:
- k is the number of times an event occurs in an interval and k can take values 0, 1, 2, ….
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
- Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur.
- The probability of an event in a small sub-interval is proportional to the length of the sub-interval.