[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذا الرابط]
Related distributions
- If [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة] and [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة] then the difference Y = X1 − X2 follows a Skellam distribution.
- If [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة] and [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة] are independent, and Y = X1 + X2, then the distribution of X1 conditional on Y = y is a binomial. Specifically, [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]. More generally, if X1, X2,..., Xn are independent Poisson random variables with parameters λ1, λ2,..., λn then [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
- The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05. According to this rule the approximation is excellent if n ≥ 100 and np ≤ 10.[1]
- For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ, and variance λ, is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., P(X ≤ x), where (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5).
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
- Variance stabilizing transformation: When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about [ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة] and variance of about 1/4.[2] Under this transformation, the convergence to normality is far faster than the untransformed variable. Other, slightly more complicated, variance stabilizing transformations are available,[3] one of which is Anscombe transform. See Data transformation (statistics) for more general uses of transformations.
- If the number of arrivals in a given time interval [0,t] follows the Poisson distribution, with mean = λt, then the lengths of the inter-arrival times follow the Exponential distribution, with mean 1 / λ.
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذا الرابط]
Occurrence
The Poisson distribution arises in connection with Poisson processes. 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 space. Examples of events that may be modelled as a Poisson distribution include:
Occurrence
The Poisson distribution arises in connection with Poisson processes. 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 space. Examples of events that may be modelled as a Poisson distribution include:
- The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
- The number of phone calls at a call centre per minute.
- Under an assumption of homogeneity, the number of times a web server is accessed per minute.
- The number of mutations in a given stretch of DNA after a certain amount of radiation.
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذا الرابط]
How does this distribution arise? — The law of rare events
In several of the above examples—for example, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution. However, the binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity. This provides a means by which to approximate random variables using the Poisson distribution rather than the more-cumbersome binomial distribution.
This limit is sometimes known as the law of rare events, since each of the individual Bernoulli events rarely triggers. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter λ is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
The proof may proceed as follows. First, recall from calculus
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
and the definition of the Binomial distribution
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
If the binomial probability can be defined such that p = λ / n, we can evaluate the limit of P as n goes large:
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
The F term can be written as
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
and then note that, since k is fixed, this is a rational function of n with limit 1.
Consequently, the limit of the distribution becomes
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
which now assumes the Poisson distribution.
More generally, whenever a sequence of independent binomial random variables with parameters n and pn is such that
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
How does this distribution arise? — The law of rare events
In several of the above examples—for example, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution. However, the binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity. This provides a means by which to approximate random variables using the Poisson distribution rather than the more-cumbersome binomial distribution.
This limit is sometimes known as the law of rare events, since each of the individual Bernoulli events rarely triggers. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter λ is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
The proof may proceed as follows. First, recall from calculus
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
and the definition of the Binomial distribution
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
If the binomial probability can be defined such that p = λ / n, we can evaluate the limit of P as n goes large:
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
The F term can be written as
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
and then note that, since k is fixed, this is a rational function of n with limit 1.
Consequently, the limit of the distribution becomes
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
which now assumes the Poisson distribution.
More generally, whenever a sequence of independent binomial random variables with parameters n and pn is such that
[ندعوك للتسجيل في المنتدى أو التعريف بنفسك لمعاينة هذه الصورة]
the sequence converges in distribution to a Poisson random variable with mean λ (see, e.g. law of rare events