Probability distributions Poisson distribution

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

The Statistics toolkit contains many useful functions for stochastic simulation. A uniform random number in [0, 1] is returned by rand randn returns a random number distributed by the normal distribution with a mean of zero and a variance of 1 (for more general, and multivariate, normal distributions, use normrnd). The normal probability distribution, cumulative distribution, and inverse cumulative probability distribution are returned by normpdf, normcdf, and norminv respectively. Similar routines are available for other distributions for example, the Poisson probability density function is returned by poisspdf. A GUI tool, df ittool, is available to fit data to a probability distribution. The mean, standard variation, and variance of a data set are returned by mean, std, and var respectively. For a more comprehensive listing of the available functions, consult the documentation for the Statistics tooikit. 

We propose to describe the distribution of the number of fronts crossing x by the Poisson distribution function, discussed in Sec. 1.9. This probability distribution function describes the probability P(F) of a specific number of fronts F in terms of that number and the average number F as follows [Eq. (1.38)] ... 

If the failure distribution of a component i.s exponential, the conditional probability of observing exactly M failures in test time t given a true (but unknown) failure rate A and a Poisson distribution, is equation 2.6-9. The continuous form of Bayes s equation is equation... 

Here, f(x) is tlie probability of x occurrences of an event tliat occurs on the average p times per unit of space or time. Both tlie mean and tlie variance of a random variable X liaving a Poisson distribution are (i. 

Assume tlie number of particles emitted by a radioactive substance has a Poisson distribution with an average emission of one particle per second, (a) Find tlie probability tliat at most one particle will be emitted in 3 seconds. 

Tlie conditional probability of event B, no failures in 10 years, given tliat tlie failure rate is Z per year, is obtained by applying the Poisson distribution to give... 

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... 

Conditional probabilities of failure can be used to predict the number of unfailed units that will fail within a specified period on each of the units. For each unit, the estimate of the conditional probability of failure within a specified period of time (8000 hours here) must be calculated. If there is a large number of units and the conditional probabilities are small, then the number of failures in that period will be approximately Poisson distributed (a special form of the normal distribution), with mean equal to the sum of the conditional probabilities, which must be expressed as decimals rather than percentages. The Poisson distribution allows us to make probability statements about the number of failures that will occur within a given period of time. 

Equation 10-6 is the well-known Poisson distribution,5 which is important in counting whenever the number of counts taken is low enough to make a count of zero fairly probable. The analytical chemist, except occasionally in trace determinations, wrill deal with counts so large that he need not concern himself with the Poisson distribution. 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

The gaussian distribution is a good example of a case where the mean and standard derivation are good measures of the center of the distribution and its spread about the center . This is indicated by an inspection of Fig. 3-3, which shows that the mean gives the location of the central peak of the density, and the standard deviation is the distance from the mean where the density has fallen to e 112 = 0.607 its peak value. Another indication that the standard deviation is a good measure of spread in this case is that 68% of the probability under the density function is located within one standard deviation of the mean. A similar discussion can be given for the Poisson distribution. The details are left as an exercise. 

There are many ways we could assign probability distribution functions to the increments N(t + sk) — N(t + tk) and simultaneously satisfy the independent increment requirement expressed by Eq. (3-237) however, if we require a few additional properties, it is possible to show that the only possible probability density assignment is the Poisson process assignment defined by Eq. (3-231). One example of such additional requirements is the following50... 

Our first objective is going to be the determination of the finite order probability density function of Y(t) in terms of the known finite order probability densities for the increments of N(t). In preparation for this, we first note that, since N(s) — N(t) is Poisson distributed with parameter.n(s — t) for s > t, it follows that... 

The number of arrivals follows a Poisson distribution when samples are submitted independently from each other, which is generally valid when the samples are submitted by several customers. The probability of n arrivals in a time interval t... 

Fig. 42.5. The distribution of the probability that n samples arrive per day, observed in a department for structural analysis. (I) observed ( ) Poisson distribution with mean p.

Similar mathematical solution can be derived from a Poisson distribution of random events in 2D space. The probability that 2D separation space will be covered by peaks in ideally orthogonal separation is analogical to an example where balls are randomly thrown in 2D space divided into uniform bins. The general relationship between the number of events K (number of balls, peaks, etc.) and the number of bins occupied F (bins containing one or more balls, peaks, etc.) is described by Equation 12.3, where N is the number of available bins (peak capacity in 2DLC). 

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. 

The next step is to generate all possible and allowed conformations, which leads to the full probability distribution F). The normalisation of this distribution gives the number of molecules of type i in conformation c, and from this it is trivial to extract the volume fraction profiles for all the molecules in the system. With these density distributions, one can subsequently compute the distribution of charges in the system. The charges should be consistent with the electrostatic potentials, according to the Poisson equation ... 

The outflow of a CSTR is a Poisson process, i.e., fluid elements are randomly selected regardless of theirposition in the reactor. The waiting time before selection for a Poisson process has an exponential probability distribution. 

Figure 24 Probability distributions for the waiting time for 10 dihedral transitions. Time is given in units of the average waiting time 10x. The distributions are peaked around 10 = 1 and are much broader than the Poisson distribution but approach it for high T. For low T, a high probability for short waiting times exists and a long time tail of the distribution develops.

Due to system complexity and paucity of information about the reaction mechanism(s), a Markov process model is proposed with probability pk = a cxpf -X)/k of the production of k ions of the anionic species X is the mean of the postulated Poisson distribution. The model also stipulates a probability q of reaction of the anionic species during two successive equal time periods of At each. 

The limiting probabilities p for the number of ionic species in the system can be established in a manner similar to the production/decomposition of a radioactive particle possessing the same probabilistic properties6 (Example 38.3, pp. 237-238). The result of an involved algebraic manipulation, itself a Poisson distribution ... 

The probability that n quenchers reside within this volume is assumed to obey a Poisson distribution ... 

The above considerations can be generalized to complexes of the type M Q ( > 1). The probability that a molecule M is in contact with n quencher molecules can be approximately expressed by the Poisson distribution (Eq. 4.21). Perrin s equation (4.23) is then found again. 

Already in the study of linear chain molecules it has become evident that the shape of the molar mass distribution and its width provide a valuable guide to the mechanism of chain formation. Best known are the most probable (or Schulz-Flory) distribution and the narrow Poisson distribution. The former is often... 

The molar mass distribution of branched materials differ most significantly from those known for Hnear chains. To make this evident the well known types of (i) Schulz-Flory, or most probable distribution, (ii) Poisson, and (iii) Schulz-Zimm distributions are reproduced. Let x denote the degree of polymerization of an x-mer. Then we have as follows. 

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