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The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution , and this leads to an alternative expression.
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion.
Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. The Conway–Maxwell–Poisson distribution, a two-parameter extension of the Poisson distribution with an adjustable rate of decay.
The limiting case n −1 = 0 is a Poisson distribution. The negative binomial distributions, (number of failures before r successes with probability p of success on each trial). The special case r = 1 is a geometric distribution. Every cumulant is just r times the corresponding
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
Figure 1: The left graph shows a probability density function. The right graph shows the cumulative distribution function. The value at a in the cumulative distribution equals the area under the probability density curve up to the point a. Absolutely continuous probability distributions can be described in several ways.
a function of t, determines the behavior and properties of the probability distribution of X. It is equivalent to a probability density function or cumulative distribution function, since knowing one of these functions allows computation of the others, but they provide different insights into the features of the random variable. In particular ...
The geometric distribution is a special case of discrete compound Poisson distribution. [ 11 ] : 606 The minimum of n {\displaystyle n} geometric random variables with parameters p 1 , … , p n {\displaystyle p_{1},\dotsc ,p_{n}} is also geometrically distributed with parameter 1 − ∏ i = 1 n ( 1 − p i ) {\displaystyle 1-\prod _{i=1}^{n ...