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In statistics and probability theory, the Poisson distribution is a discrete probability distribution (discovered by Siméon-Denis Poisson ( 1781- 1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile) belonging to certain random variables N that count, among other things, a number of discrete occurrences (sometimes called "arrivals") that take place during a time-interval of given length. The probability that there are exactly k occurrences (k being a natural number including 0, k = 0, 1, 2, ...) is:-
Where:
- is the base of the natural logarithm ( = 2.71828...),
- is the factorial of ,
- is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 2 minutes, and you are interested in the number of events occcurring in a 10 minute interval, you would use as model a Poisson distribution with .
1 Poisson processes
Sometimes is taken to be the rate, i.e., the average number of occurrences per unit time. In that case, if Nt is the number of occurrences before time t then we have
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and the waiting time T until the first occurrence is a continuous random variable with an exponential distributionIn probability theory and statistics, the exponential distribution is a continuous probability distribution. Specification of the exponential distribution Probability density function The probability density function (pdf) of the Exponential lambda distri; this probability distribution may be deduced from the fact that
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When time becomes involved, then we have a 1-dimensional Poisson processA Poisson process one of a variety of things named after the French mathematician Simeon-Denis Poisson (1781 1840), is a stochastic process that assigns to each bounded interval of time or to each bounded region in some space (for example, a Euclidean pla, which involves both the discrete Poisson-distributed random variables that count the number of arrivals in each time interval, and the continuous Erlang-distributedThe Erlang distribution is a probability distribution developed by A. Erlang to predict waiting times in queuing systems, particularly in the case of telephone traffic engineering. The Erlang distribution is the distribution of the sum of independent iden waiting times. There are also Poisson processA Poisson process one of a variety of things named after the French mathematician Simeon-Denis Poisson (1781 1840), is a stochastic process that assigns to each bounded interval of time or to each bounded region in some space (for example, a Euclidean plaes of dimension higher than 1.
2 Occurrence
The Poisson distribution arises in connection with Poisson processA Poisson process one of a variety of things named after the French mathematician Simeon-Denis Poisson (1781 1840), is a stochastic process that assigns to each bounded interval of time or to each bounded region in some space (for example, a Euclidean plaes. It applies to various phenomena of discrete nature (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or spaceThe word space has many meanings, including: Physics The definition of space in physics is contentious. Various concepts used to try to define space have included: the structure defined by the set of "spatial relationships" between objects a manifold defi. Examples include:
- The number of unstable nucleiThe nucleus atomic nucleus is the center of an atom. It is composed of one or more protons and usually some neutrons as well. The number of protons in an atom's nucleus is called the atomic number, and determines which element the atom is (for example hyd that decayed within a given period of time in a piece of radioactive substanceRadioactive decay Radioactivity is the process by which unstable atomic nuclei decay. This process normally produces ionizing radiation with a relatively large amount of energy. This energy can be harnessed in the form of nuclear power, or it can be very.
- The number of cars that pass through a certain point on a road during a given period of time.
- The number of spelling mistakes a secretary makes while typing a single page.
- The number of phone calls at a call center per minute.
- The number of times a web server is accessed per minute.
- For instance, the number of edits per hour recorded on Wikipedia's page follows an approximately Poisson distribution.
- The number of roadkill found per unit length of road.
- The number of mutations in a given stretch of DNA after a certain amount of radiation.
- The number of pine trees per unit area of mixed forest.
- The number of stars in a given volume of space.
- The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry (an example made famous by a book of Ladislaus Josephovich Bortkiewicz ( 1868- 1931)).
- The number of bombs falling on each unit area of London during a German air raid in the early part of the Second World War.
- The distribution of visual receptor cells in the retina of the human eye.
