Poisson distribution, discrete probability

Poisson distribution A probability distribution for a discrete random variable. It is defined, for a variable (r) that can take values in the range 0,1,2,. and has a mean value p, as... 

The Poisson distribution can be used to determine probabilities for discrete random variables where the random variable is the number of times that an event occurs in a single trial (unit of lime, space, etc.). The probability function for a Poisson random variable is... 

In the Jirst example a customer orders 1 unit with 70% probability and 5 units with 30% probability. The number of orders per period is Poisson distributed with mean 4. Figure 6.2 shows the resulting (discrete) compound Poisson density and the cumulated distribution and their gamma approximations. 

These apply to discrete characteristics which can assume low whole-number values, such as counts of events occurring in area, volume or time. The events should be rare in that the mean number observed should be a small proportion of the total that could possibly be found. Also, finding one count should not influence the probability of finding another. The shape of Poisson distributions is described by only one parameter, the mean number of events observed, and has the special characteristic that the variance is equal to the mean. The shape has a pronounced positive skewness at low mean counts, but becomes more and more symmetrical as the mean number of counts increases (Fig. 41.3). 

Following are some examples of frequently encountered probability distributions Poisson distribution. This is the discrete distribution... 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

The Poisson distribution describes the probability of a discrete number of events occurring within a fixed interval, given that the probability of the event occurring is independent of the size of the interval. For example, suppose that a manufacturing defect occurs with an average rate of occurrence p and the products are manufactured over an interval n. The expected number of defects is clearly np. The Poisson distribution is one way of describing the probability that x defects will occur in an interval n. This distribution is a generalization of the binomial distribution to an infinite number of trials. The mathematical form of the Poisson distribution [2] is... 

Probability mass function. A function which assigns a probability to values of a random variable. Used for discrete variables as opposed to probability density function, which is used for continuous ones. An example is given in the entry under Poisson distribution. 

The Poisson distribution is based on the probability density function for discrete values of a variate. This is termed a probability function. For each value of this function,/(. ), a probability for the realization of the event, x, can be defined. It is calculated according to a Poisson distribution by... 

Discrete probability distributions include the Binomial distribution and the Poisson distribution. 

Consider the problem of updating the occurrence rate of an event, such as a certain level of earthquakes or typhoons in a particular region. The discrete Poisson distribution is a well-known probabilistic model for this purpose and the probability of exactly k(>0) occurrences of an event in a specified time interval is given by ... 

We note the Poisson distribution holds true for discrete values of a. Speaking in terms of macromolecules, Equation (20) yields the probability for attaching N - 1 monomers to a given monomer. Hence, the number fraction of chains having... 

Table 6. Poisson Distribution. This is a discrete distribution approximating the binomial when the total number of items of data (the populations) is very large, but the probability (p) is very small and the sample is small compared with the population...

Poisson Distribution n A probability distribution of a discrete random variable, X, with non-negative integer values, X, that have a probability, P(x), of the form ... 

Gaussian and Poisson distributions are related in that they are extreme forms of the Binomial distribution. The binomial distribution describes the probability distribution for any number of discrete trials. A Gaussian distribution is therefore used when the probability of an event is large (this results in more symmetric bell-shaped curves), whereas a Poisson distribution is used when the probability is small (this results in asymmetric curves). The Lorentzian distribution represents... 

The Poisson distribution (pronounced [pwas5]) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. (Poisson distribution can also be used for the number of events in other specified intervals such as distance, area, or volume.)... 

In order to determine a for uncorrelated data, we only need to calculate the probability PQii) that in the rth thread a state with energy E (for simphcity we assume that the problem is discrete) is hit /j, times in M, trials, where each hit occurs with the probability hit This leads to the binomial distribution with the hit average hi) = Mjphit- In the limit of small hit probabilities (a reasonable assumption in general if the number of energy bins is large, and, in particular, for the tails of the histogram), the binomial turns into the Poisson distribution P hi) /hi with identical variance and expectation value, a, = hi). Insertion... 

It is helpful to have standard probabihty models that are useful for analyzing large biological data, in particular bioinformatics. There are six standard distributions for discrete r.v. s, that is, BemouUi for binary r.v. s, (e.g., success or failure), binomial for the number of successes in n independent BemouUi trials with a common success probabihty p, uniform for model simations where aU integer outcomes have the same probabihty over an interval [a, b, geometric for the number of trials required to obtain the first success in a sequence of independent BemouUi trials with a common success probabihty p, Poisson used to model the number of occurrences of rare events, and negative binomial for the number of successes in a fixed number of Bemoulh trials, each with a probability p of success. 

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