# Poisson Probability Distribution Family

A Poisson Probability Distribution Family, [math]\displaystyle{ \mathrm{Pois}(\lambda) }[/math], is a discrete probability distribution family characterized by the Poisson 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 a random event, and [math]\displaystyle{ \lambda }[/math] is a positive real number representing the expected number of occurrences during a given interval.

**Context:**- It can (typically) be parameterized solely by its Mean which is equal to its Variance.
- It can be instantiated as a Poisson Probability Density Function, Poisson Probability Mass Function.
- It can be used to model events that occur randomly in both time and space, thereby often forming the basis of a Poisson Process.
- It can serve as an approximation to the Binomial Distribution when the number of trials is large and the probability of success in each trial is small.
- It can be generalized into Compound Poisson Distributions.
- ...

**Example(s):**- A dataset {(0, 0.674%), (1, 3.369%), (2, 8.422%), (3, 14.037%), (4, 17.547%), (5, 17.547%), (6, 14.622%), (7, 10.444%), (8, 6.528%), (9, 3.627%), (10, 1.813%), (11, 0.824%), (12, 0.343%), (13, 0.132%), (14, 0.047%), (15, 0.016%), (16, 0.005%)}, representing a Poisson Probability Distribution with an average of 5 events per unit time or space. Source: [GraphPad](http://www.graphpad.com/quickcalcs/probability1.cfm)
- …
- The number of phone calls received by a call center in an hour.
- The number of mutations in a given stretch of DNA after a certain amount of radiation exposure.
- The number of cars passing through a certain point in a given time period.
- ...

**Counter-Example(s):**- An Exponential Distribution, , such as a Gaussian distribution used for modeling the time between events in a Poisson Process.
- A Binomial Distribution, which is used when there are only two possible outcomes for each trial and each trial is independent.
- A Gamma Distribution, which can be used to model the waiting time until the nth event in a Poisson Process.
- A Log-Normal Distribution, which is used for modeling variables that can take on any positive value and are subject to multiplicative effects.

**See:**Cumulative Poisson Probability Distribution, Poisson Random Variable, Poisson Regression, Poisson Mean Value, Poisson Mixture Model, Negative Binomial Mass Function.

## References

### 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