Poisson probability distribution

The Poisson probability distribution is named after the French mathematician Simeon D. Poisson.

Suppose there is a power outage in an apartment complex 3 times a year.

You may want to find the probability that there will be exactly 2 power outage next year. This is an example of a Poisson probability distribution problem.

Each power outage is called an occurrence. In order to use the Poisson probability distribution, the occurrence must be independent and random.

In our example of power outage, each power outage is random. In other words, they do not follow any known pattern.

There is no way to tell when a power outage will occur. They just happen randomly.

The power outage is also independent. The next power outage does not depend on the one before. In other words, a power outage this year does not influence a power outage next year.

Occurrences always happen with respect to an interval. In the example of power outage, the interval in one year and this make sense since we do not expect too many power outage unless we are living in a less developed country.

Once you know the average number of occurrences in an interval, we use the Poisson probability distribution to compute the probability of a certain number of occurrences ,x, in that interval.

Examples to which the Poisson probability distribution can be applied

  • The number of patients arriving at the emergency room during a one-hour interval.
  • The number of phone calls you will receive from telemarketers during a one-month interval.
  • The number of defective items in the next 200 items produced by a machine. 
  • The number of vehicle accidents that occur on a given highway during a one-week interval.
Notice that the last example is an example of volume interval instead of time interval. The occurrences are the number of defective items. They are random since there could be 0, 1, ... , 100 defective items. These occurrences are random since the occurrence of a defective item does not influence the occurrence of another.

Poisson probability distribution formula

The probability of x occurrences in an interval is 

$$ P(x)\ =\ \frac{λ^x e^-λ}{x!} $$

λ is called parameter of the Poisson probability distribution or Poisson parameter. λ is the mean number of occurrences in that interval and is pronounced lambda. The value of e is 2.71828.

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