Poisson distribution: Difference between revisions
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The '''Poisson distribution''' is any member of a class of [[discrete probability distribution|discrete probability distributions]] named after [[Simeon Denis Poisson]]. | The '''Poisson distribution''' is any member of a class of [[discrete probability distribution|discrete probability distributions]] named after [[Simeon Denis Poisson]]. | ||
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===Example=== | ===Example=== | ||
Certain events happen at unpredictable intervals. But for some reason, no matter how recent or long ago last event was, the probability that another event will occur within the next hour is exactly the same (say, 10%). The same holds for any other time interval (say, second). Moreover, the number of events within any given time interval is statistically independent of numbers of events in other intervals that do not overlap the given interval. Also, two events never occur simultaneously. | |||
Then the number of events per day is Poisson distributed. | Then the number of events per day is Poisson distributed. | ||
===Formal definition=== | ===Formal definition=== | ||
Let X be a stochastic variable taking non-negative integer values with [[probability density function]] <math>P(X=k)=f(k)= e^{-\lambda} \frac{\lambda ^k}{k!} </math> | Let X be a stochastic variable taking non-negative integer values with [[probability density function]] | ||
: <math> P(X=k)=f(k)= e^{-\lambda} \frac{\lambda ^k}{k!}. </math> | |||
Then X follows the Poisson distribution with parameter <math>\lambda</math>. | |||
===Characteristics of the Poisson distribution=== | ===Characteristics of the Poisson distribution=== | ||
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*The [[variance]] <math>Var[X]=\lambda</math> | *The [[variance]] <math>Var[X]=\lambda</math> | ||
<!-- *The entropy <math>H=</math> --> | <!-- *The entropy <math>H=</math> --> | ||
==See also== | ==See also== | ||
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*[[Probability theory]] | *[[Probability theory]] | ||
== | ==References== | ||
{{reflist}} | |||
Latest revision as of 09:15, 14 September 2013
The Poisson distribution is any member of a class of discrete probability distributions named after Simeon Denis Poisson.
It is well suited for modeling various physical phenomena.
A basic introduction to the concept
Example
Certain events happen at unpredictable intervals. But for some reason, no matter how recent or long ago last event was, the probability that another event will occur within the next hour is exactly the same (say, 10%). The same holds for any other time interval (say, second). Moreover, the number of events within any given time interval is statistically independent of numbers of events in other intervals that do not overlap the given interval. Also, two events never occur simultaneously.
Then the number of events per day is Poisson distributed.
Formal definition
Let X be a stochastic variable taking non-negative integer values with probability density function
Then X follows the Poisson distribution with parameter .
Characteristics of the Poisson distribution
If X is a Poisson distribution stochastic variable with parameter , then
- The expected value
- The variance
![{\displaystyle E[X]=\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c0f342837deda84cd6059ba07fc858af2d9400)
![{\displaystyle Var[X]=\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/203df9f1c2718876a191df085fa3ad610711af68)
See also
- Binomial distribution
- Exponential distribution
- Probability distribution
- Probability
- Probability theory