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The probability distributions of Poisson and negative exponential They are frequently used in problems of queuing theory; the Poisson distribution represents a number of independent events occurring at a constant speed in a time interval specific expressed in units ranging for example from seconds even years. It allows modeling situations as diverse as the number of calls arriving at a call center the number of bacteria reproduced in a certain population the time that can last a path closed because of a landslide the number of people arriving to a self insurance or the amount requested an insurer; the negative exponential distribution represents the length of the intervals' occurrence between events that are distributed according to the distribution Poisson (Canvass). The Poisson process relates the Poisson and exponential refusal to treat problems of the theory tails.
Then an overview of each of these steps forward distributions and concepts and models included in this theory the order to advance the review of specific situations of health management allowing your application. X is a random variable representing independent random events that occur at a constant speed over time or space. It is said that the random variable X has a Poisson distribution with queue management following probability distribution (Cantos): It is said in this way that the random variable X has a distribution Poisson with parameter (Meyer). For details of the deduction of formula Poisson see Canvass (). The random variables HT and Yo They are both defined by the same probability distribution this is known as the best value probability and depends only on the size of the interval in which it is deemed and not the end points of the range. Mathematically it is said that the probability function that defines events that occur in the interval Interest has stationary independent increments.
The above definition corresponds to a probability distribution since it satisfies the following axioms of a probability space. The Poisson probability distribution has the sample space the set of integers.. numbers; the probability associated with each replacing sample point is obtained directly in the distribution formula probability presented. The function is always positive because each of the three factors that define is also positive for any value of k according to the following arguments: an exponential function is always positive. Parameter means for any k and finally queue management software the factorial of a positive integer is always greater or equal to one k! ≥. b. The values that can take the random variable time intervals non overlapping they are considered independent random variables. Two or more events A and B are independent if P (A ∩ B) P (A) P (B) result which can be generalized for two or more independent events. A. Let The random queue management software variable defined as above and is YT Another variable Random defining the number of events that occur in the interval (t+ t) for any t.
Developments using power series and applying properties the sum over all values of the sample space we have: Where E (X) denotes the expected value of the random variable this flag also known as average and VAR (X) denotes the variance of the random variable. The Poisson distribution has the property interesting to have their (average) equal to its expected value Variant One of continuous distributions which has found application in a broad class of phenomena characteristic of queuing theory and the theory of Reliability is called negative exponential distribution a particular case gamma distribution and Rang distribution; this function is completely determined by a queue. The function defining parameter This probability distribution is important The following characteristics to identify the Poisson model as the ideal to a characterization of an experiment of queuing theory (Meyer).
It is a constant given if p (At) is approximately equal to view more where CDT is small enough. This is expressed as: This queue management solution means that if the interval is sufficiently small the probability associated to the occurrence of an event is directly proportional to said interval length. At is a random variable (stochastic process) that can take the values.. during a time interval (t) and Denote the probability that the random variable takes the value n at time t as follows: The random variable queue management solution must meet certain conditions in order to conclude that follows a Poisson model. Some of these considerations They must be accepted as true in order to formulate a model Poisson; mathematical rigor can be found in texts