Poisson distributions

A Poisson distribution occurs when a constant number of polymer chains begin to grow simultaneously and when the addition of monomeric units is random and occurs independently of the previous addition of other monomeric units. Consequently, Poisson distributions may occur with what are called living polymers (see Sections 15 and 18). 

8 Molar Masses and Molar Mass Distributions 

For the differential molar degree of polymerization distribution, the Poisson distribution gives 

Consequently, the ratio XwjXn in the Poisson distribution depends on the number-average degree of polymerization and on no other parameter. The ratio XwjXn tends to a value of unity with increasing degree of polymerization. The Poisson distribution is consequently a very narrow distribution. 

The Poisson distribution describes the results of experiments in which we count events that occur at random but at a definite average rate. Examples of the Poisson distribution include the number of emails we receive in a one-day period, the number of babies bom in a hospital in a two-day period, the number of decays of a radioactive isotope in a one-day period. 

Example. Suppose we know that we receive five emails (excluding junk emails) as an average in a one-day period, what is the probability that we receive three emails in a specific day  

Example. For the same average (5 emails/day), what is the chance of receiving 10 emails in a specific day  

The Poisson distribution has important applications in isotope geochemistry where counting statistics is needed. One example in 

The total number of events N is large and x is small. For the first approximation 

The probability of success p is small. Hence, the factor (1 — is nearly equal to unity, (1 — pY = 1. 

Take the average value of m (the number of successfiil events) from the binomial distribution  

When we apply al three of these conditions, Eq. (3.1) becomes 

Equation (3.2) represents the Poisson distribution and/(x) is the Poisson distribution function. Like binomial distribution, Poisson distribution is not continuous. To calculate the average (mean) value of x, we must use the operator summation. The value of m that is the product of Np plays a characteristic role in Poisson distribution. An important assumption is that the distribution f x) is through the area with uniform probability. In comparison, the binomial distribution has a much wider spread from the lowest value to the highest value than the Poisson distribution. 

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Figure Bl.10.1. Poisson distribution for fi(left vertical axis). Cumulative Poisson distribution for fi= l(right vertical axis). The cumulative distribution is the sum of the values of the distribution from 0 to n, where n =, 2, 3, 4, 5 on this graph.

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... 

Figure 10. Level spaeitig distributions P s/ s)) for the cone states of the first-excited electronic doublet state of Li3 with consideration of GP effects [12] (a) Ai symmetry (b) A2 symmetry (c) E symmetry (d) full spectrum. Also shown by the solid lines are the corresponding fits to a Poisson distribution.

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... 

Eig. 1. Ziegler ethylene chain growth. Theoretical (Poisson) distribution of primary alcohols at ( ) 2.5, (- -... 

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 ... 

Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation ... 

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). 

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... 

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... 

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. 

In addition to tlie applications cited above, tlie Poisson distribution can be used to obtain tlie reliability of a standby redundancy system, in which one unit is in tlie operating mode and n identical units are in standby mode. Unlike a... 

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. 

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. 

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