# Poisson Probability Function

A Poisson Probability Function is a discrete probability function, [math]\displaystyle{ \mathrm{Pois}(\lambda, k) = \frac{\lambda^k e^{-\lambda}}{k!} }[/math], where [math]\displaystyle{ k }[/math] is the number of occurrences of an random event, and [math]\displaystyle{ \lambda }[/math] is a positive real number equal to the expected number of occurrences during the given interval.

**AKA:**Poisson PDF.**Context:**- It can (typically) be a member of a Poisson Probability Distribution Family.
- It can be used to model a Poisson Process, where:
- events can be counted in whole numbers;
- occurrences are independent (do not impact the chances of a future event).
- the average frequency of occurrence for the time period in question is known; and
- it is possible to count how many events have occurred, such as the number of times a firefly lights up in my garden in a given 5 seconds, some evening, but meaningless to ask how many such events have not occurred.

- …

**Example(s):****Counter-Example(s):**- an Exponential Probability Function, such as a Gaussian distribution.

**See:**Poisson Mixture Model, Negative Binomial Mass Function.

## References

### 2016

- https://www.umass.edu/wsp/resources/poisson/
- QUOTE: The Poisson distribution applies when:
- the event is something that can be counted in whole numbers;
- occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another;
- the average frequency of occurrence for the time period in question is known; and
- it is possible to count how many events have occurred, such as the number of times a firefly lights up in my garden in a given 5 seconds, some evening, but meaningless to ask how many such events have not occurred.

- This last point sums up the contrast with the Binomial situation, where the probability of each of two mutually exclusive events (p and q) is known. The Poisson Distribution, so to speak, is the Binomial Distribution Without Q. In those circumstances, and they are surprisingly common, the Poisson Distribution gives the expected frequency profile for events. It may be used in reverse, to test whether a given data set was generated by a random process. If the data fit the Poisson Expectation closely, then there is no strong reason to believe that something other than random occurrence is at work. On the other hand, if the data are lumpy, we look for what might be causing the lump.

- QUOTE: The Poisson distribution applies when:

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Poisson_distribution Retrieved:2015-6-14.
- In probability theory and statistics, the
**Poisson distribution**, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail doesn't affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received per day obeys a Poisson distribution. Other examples that may follow a Poisson: the number of phone calls received by a call center per hour, the number of decay events per second from a radioactive source, or the number of taxis passing a particular street corner per hour.

- In probability theory and statistics, the

### 2011

- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Poisson_distribution
- … If the expected number of occurrences in this interval is
*λ*, then the probability that there are exactly [math]\displaystyle{ k }[/math] occurrences (*k*being a non-negative integer, [math]\displaystyle{ k }[/math] = 0, 1, 2, ...) is equal to [math]\displaystyle{ f(k, \lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\! }[/math], where- [math]\displaystyle{ e }[/math] is the base of the natural logarithm (
*e*= 2.71828...) - [math]\displaystyle{ k }[/math] is the number of occurrences of an event — the probability of which is given by the function
*k*! is the factorial of*k**λ*is a positive real number, equal to the expected number of occurrences during the given interval.

- [math]\displaystyle{ e }[/math] is the base of the natural logarithm (
- For instance, if the events occur on average 4 times per minute, and one is interested in the probability of an event occurring k times in a 10 minute interval, one would use a Poisson distribution as the model with λ = 10×4 = 40.
- As a function of [math]\displaystyle{ k }[/math], this is the probability mass function. The Poisson distribution can be derived as a limiting case of the binomial distribution.
- The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms.

- … If the expected number of occurrences in this interval is

### 2009

- (MathWorks, 2009) ⇒ http://www.mathworks.com/help/toolbox/stats/poisspdf.html
- Syntax: Y = poisspdf(X,lambda)
- Description: Y = poisspdf(X,lambda) computes the Poisson pdf at each of the values in X using mean parameters in lambda. X and lambda can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other input. The parameters in lambda must all be positive.
- The Poisson pdf is [math]\displaystyle{ f(x, \lambda)=\frac{\lambda^x}{x!}e^{-\lambda},\,\! I_{(0,1,...)}(x) }[/math] where x can be any nonnegative integer. The density function is zero unless x is an integer.

Examples

- QUOTE: A computer hard disk manufacturer has observed that flaws occur randomly in the manufacturing process at the average rate of two flaws in a 4 GB hard disk and has found this rate to be acceptable. What is the probability that a disk will be manufactured with no defects? In this problem, λ = 2 and x = 0. p = poisspdf(0,2) p = 0.1353