The Poisson Distribution

The Poisson distribution applies to events whose probability of occurrence is small and constant. It can be derived from the binomial distribution by letting 

The Poisson distribution has wide applications in many diverse fields, such as decay of nuclei, persons killed by lightning, number of telephone calls received in a switchboard, emission of photons by excited nuclei, and appearance of cosmic rays. 

Example 2.10 A radiation detector is used to count the particles emitted by a radioisotopic source. If it is known that the average counting rate is 20 counts/ min, what is the probability that the next trial will give 18 counts/min  

Answer The probability of decay of radioactive atoms follows the Poisson distribution. Therefore, using Eq. 2.50, 

That is, if one performs 10,000 measurements, 844 of them are expected to give the result 18 counts/min. 

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When the rate measurement is statistically distributed about the mean, the distribution of events can be described by the Poisson distribution, Prrf O, given by... 

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)] ... 

That the Poisson distribution results in a narrower distribution of molecular weights than is obtained with termination is shown by Fig. 6.11. Here N /N is plotted as a function of n for F= 50, for living polymers as given by Eq. (6.109). and for conventional free-radical polymerization as given by Eq. (6.77). This same point is made by considering the ratio M /M for the case of living polymers. This ratio may be shown to equal... 

Nature In monitoring a moving threadhne, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadhne by a broken-filament detector mounted adjacent to the threadhne. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include ... 

The expectation numbers were computed as follows For the Poisson distribution, X = E(x) therefore, an estimate of X is the average number of defective units per worker, i.e., X = (l/52)(0 x3-l-lx7-l---1-9x1) = 3.23. Given this... 

As X increases, the Poisson distribution approaches the normal distribution, with the relationship... 

When the value of p is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with X = np when n > 50 while npi < 5. 

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

The Poisson distribution follows naturally from the discrete binomial distribution already introduced in the craps and the M-out-of-N problem. As N becomes large, the Poisson distribution approximates the binomial distribution... 

The Poisson distribution for observing M events in time r is given by equation 2.5-1, where / is the failure rate estimated as M/i. This model may be used if the failure rate is time dependent rather than demand... 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

Tlie pdf of the Poisson distribution can be derived by taking the limit of tlie binomial pdf as n qo, p 0, and np = remains constant. The Poisson pdf is given by... 

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. 

What distribution does concern him It turns out to be the Gaussian, for the Poisson distribution as N becomes larger approaches more and more closely to the Gaussian. It may be shown analytically that, for large N,... 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

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... 

A simple way to calculate the variance of the Poisson distribution is to first calculate < % and then apply Eq. (3-58)... 

Both the mean and variance of the Poisson distribution are equal to the parameter A. 

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. 

We conclude this section with a derivation of the characteristic function of the Poisson distribution. Starting from the definition, Eq. (3-77), we obtain... 

A combination of continuum transport theory and the Poisson distribution of solution charges has been popular in interpreting transport of ions or conductivity of electrolytes. Assuming zero gradient in pressure and concentration of other species, the flux of an ion depends on the concentration gradient, the electrical potential gradient, and a convection... 

Eq. (32) may be deduced from kinetic arguments.This is unnecessary, however, since the conventional derivation of the Poisson distribution proceeds from equivalent statistical conditions. 

Number of monomers combined per growing center in a system yielding the Poisson distribution (Chap. VIII). 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

The validity of the Poisson distribution for silver nucleation is demonstrated in Fig. 5.48B. The assumption for this kind of treatment is that the nucleus formation is irreversible and that the event is binary consisting of a discontinuous process (nucleus formation) and a continuous process (flow of... 

Note that because die product np remains finite, the second factor in Eq. (12) approaches unity in the limit Similarly, from the definition of the exponential (Section 1.4), lim ooU - (np/n)]a = e np. Equation (13) is an expresstoh of the Poisson distribution. 

Of interest for analytical chemistry are at least two further distributions, the logarithmic normal distribution for analytical results at the trace- and ultra-trace level, and the Poisson distribution for discrete results (e.g., counts of impulse summator in XRF). 

The use of the Poisson distribution for this purpose predates the statistical overlap theory of Davis and Giddings (1983), which also utilized this approach, by 9 years. Connors work seems to be largely forgotten because it is based on 2DTLC that doesn t have the resolving power (i.e., efficiency or the number of theoretical plates) needed for complex bioseparations. However, Martin et al. (1986) offered a more modem and rigorous theoretical approach to this problem that was further clarified recently (Davis and Blumberg, 2005) with computer simulation techniques. Clearly, the concept and mathematical approach used by Connors were established ahead of its time. 

Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

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