# Probability Distribution- Poisson

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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:

1. k is the number of times an event occurs in an interval and k can take values 0, 1, 2, ….
2. The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
3. The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
4. 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.
5. The probability of an event in a small sub-interval is proportional to the length of the sub-interval.

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