Poisson probability

Poisson probability is one of the most useful probabilities available to the safety professional. A Poisson probability is used to determine the probability of an event when 

The Poisson probability formula uses an expected number of mishaps based upon previous experience (Hays 1988,144-45). 

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The Poisson pdf can be used to approximate probabilities obtained from tlie binomial pdf given in Eq. (20.5.2) when n is large and p is small. In general, good approximations will result when ii exceeds 10 and p is less tluin 0.05. When n e. cecds 100 and np is less tluui 10, tlie approximation will generally be excellent. Table 20.5.1 compares tlie binomial and Poisson probabilities of tlie case of n = 20, p =0.05 and n = 100, p = 0.01. 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

As an example, one may consider that a random process occurs where a protein may bind at a given location within the cell as often as y = 1, 2, 3,. .., times per second, but on average the protein binds once per second. If y is described by a Poisson probability function, estimate the probabihty that the protein will not bind at the site during a one-second interval. In this case, the time period is one second and the mean binding steps is A equal to 1. Therefore,... 

When n exceeds 100 and np is less tlian 10, tlie approximation will generally be excellent. Table 20.5.1 compares tlie binomial and Poisson probabilities of tlie case of n = 20, p =0.05 and n = 100, p = 0.01. 

Consider the 10-mL portions (dilution) inoculated into the five replicate tubes. If a is the expected number of bacteria per mL, the expected number of bacteria in the 10 mL (in one tube) is 2, = 10a. Thus, from the Poisson probability, for any one tube, the probability that there will be no bacteria (X = y = 0) is... 

The correction for primary coincidence is derived through Poisson probabilities of finding multiple particles in different parts of the sensing zone simultaneously. This yields the following relationship between true count N and observed count n ... 

Figure 10.2. Poisson probability functions with Gaussian approximations.

A very interesting case, not yet fully clarified, concerns the simultaneous entrapment of several (>80) macromolecular compounds (the whole transcription-translation machinery) inside submicrometric lipid vesicles (radius 100nm). ° In fact, under the conditions of the experiment, the Poisson probability to find a small vesicles containing more than 80 different compounds is 10 , i.e., critically close to zero. However, the experimental results indicate a low but well measurable yield of protein produced by the entrapped molecular machinery. Now, the calculated cumulative probability for the entrapment of ca. 80 molecules in a vesicle should be the product of 80 independent... 

So far we have a simple synthesis model, but we still lack complete techniques for actually analyzing walking sounds, specifically to determine N—the Poisson probability constant. N is estimated by inspecting a high frequency band (5.5-11 kHz) of the whitened footstep sounds. A Daubechies four-wavelet filterbank is used to split the signal into subbands, and these subbands are rectified (absolute value). As can be seen by inspecting the top... 

The random arrival of x-ray photons at the detector is described by the Poisson probability density function. More specifically, the probability of observing n photons arriving at the detector in the arbitrary time interval t is... 

Figure 4.41 shows the Poisson probability density function Pn(t) for the case fjL = 3.3. Although the envelope of the function has been drawn as a histogram, dots have been added to emphasize that the function exists only at positive integer values of n. Note that the Poisson probability density function is quite asymmetric about its mean for low values of p.. For large values of p the distribution becomes more symmetric, as will be shown in Fig. 4.43. 

Figure 4.41 The Poisson probability density function for fi = 3.3. (Reprinted by courtesy of EG G ORTEC.)...

The Normal Approximation to the Poisson Probability Density Function... 

The central limit theorem [42] shows that the normal or gaussian probability density function is a good approximation to the Poisson probability density function for large values of /u. That is,... 

Equation (4.70) shows that N is also distributed with a Poisson probability density function whose mean is given by... 

The function PN=ni-n2 is not a Poisson probability density function, nor is it particularly simple in analytical form. Consequently, another method must be used to compute the most probable value of the population mean fifi and its variance... 

More meaningful results can be obtained by assuming both ni and n2 are large compared to 30, and the gaussian approximation to the Poisson probability density function is valid. In this case Eq. (4.88) becomes... 

The photons arriving at the detector must be distributed according to the Poisson probability density function. 

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