This online Poisson Distribution Calculator computes the probability of an exact number of Poisson event occurrences (a Poisson probability P), given the number of occurrences k and the average rate of occurrences λ. You can also compute cumulative Poisson probabilities P for no more than k occurrences or for no less than k occurrences.
Poisson Distribution Formula
The 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.
According to the Poisson probability mass function, the Poisson probability of \(k\) occurrences is given by the following formula:
$$P(k,\lambda)={\frac {\lambda ^{k}e^{-\lambda }}{k!}},$$
where \(λ\) is the average rate of occurrences.
A cumulative Poisson probability refers to the probability that the Poisson experiment outcomes fall within a specified range. Thus, the lower cumulative Poisson probability is the probability of no more than k occurrences:
$${ P }_{ l }(k,\lambda)=\sum _{ x=0 }^{ k }{\frac {\lambda ^{x}e^{-\lambda }}{x!}} ,$$
and the upper cumulative Poisson probability is the probability of no less than k occurrences:
$${ P }_{ u }(k,\lambda)=\sum _{ x=k }^{ \infty }{\frac {\lambda ^{x}e^{-\lambda }}{x!}}=1-{ P }_{ l }(k,\lambda).$$
Examples of Poisson Distribution
For a better understanding of the above, we will share two examples of how the Poisson distribution is used in the real world.
Example 1: Number of Calls per Hour at a Call Center
The concept of Poisson distribution is widely used by call centers to estimate the number of employees needed to be hired.
For example, suppose that some call center receives an average of 15 calls per hour. The calls are independent, so that receiving one does not change the probability of when the next one will come. Let’s further assume that the center’s employees have long conversations with customers, mostly lasting about an hour. How many employees would you need to hire so that at least 90% of the incoming calls are answered?
Let’s use our Poisson distribution calculator to answer this question. In doing so, we should obviously use the lower cumulative Poisson probability option.
We enter the value of the average number of calls per hour into the appropriate field of the calculator: λ = 15. As the number of calls received in an hour (k) we will enter numbers starting from 15 and above. As the Poisson probability type we will choose “No more than “k” occurrences”.
Choosing different k we get the following lower cumulative probabilities:
Pl(15, 15) = 56.8%
Pl(16, 15) = 66.4%
Pl(17, 15) = 74.9%
Pl(18, 15) = 81.9%
Pl(19, 15) = 87.5%
Pl(20, 15) = 91.7%
Thus, we see that 20 is a sufficient number of employees to ensure that at least 90% of the calls coming into this call center are answered.
Example 2: Number of Arrivals at a Restaurant
Restaurants use the Poisson distribution to estimate the number of expected customers that will come to the restaurant per day.
For example, suppose a given restaurant receives an average of 120 customers per day. We can use our Poisson distribution calculator to find the probability that this restaurant will have more than a certain number of customers on any given day.
To this end, we must obviously use the upper cumulative probability:
Pu(125, 120) = 33.6%
Pu(130, 120) = 19.2%
Pu(135, 120) = 9.5%
Pu(140, 120) = 4.0%
In other words, for example, the probability that more than 140 people will visit the restaurant on a given day is only 4%.
This information gives restaurant managers an idea of the likelihood that they will get more than the usual number of customers on a given day. And this, in turn, can help them reserve resources for contingencies.
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