Poisson probability distribution

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 ... 

Predicting the rate of mutation based on the Poisson probability distribution function. The evolutionary process of amino acid substitutions in proteins is... 

Use the Poisson probability distribution to find the number of times you would expect heads (or tails) to occur in 100 coin tosses. 

When both m and Es tend to infinity ( m and Es oo), while maintaining the constant finite ratio D, the binomial distribution transforms into the Poisson probability distribution ... 

Evans [2] calculated the expectancy of the Poisson probability distribution for the constant propagation rate of domains and two simple nucleation modes instantaneous and spontaneous with the constant rate, F i) = B. Billon et al. [13] extended this approach to the case of time-dependent nucleation rate. According to the Evans theory, an arbitrarily chosen point A can be reached before time t by growing spheres nucleated around it in a distance r (precisely in a distance within the interval (r, r + dr)) before time t - rIG their number is equal to an integral of the nucleation rate F(t) over the time interval (0, t - r/G), multiplied by the considered volume, Artr dr. The total number of spheres occluding the point A until time t is calculated by second integration, over a distance ... 

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)] ... 

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... 

Figure 13 shows the relationship between the time interval At of passive film breakdown of stainless steel with chloride ions and the logarithms of cumulative probability P(Af) for breakdown at time intervals longer than At. From these results, it is clear that the logarithm of the probability is almost proportional to the time interval, and therefore the cumulative probability for film breakdown follows Poisson s distribution, i.e., the following equation is obtained,... 

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.

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. 

Fig. 19. Weight fraction molar mass distributions w(x) of the Schulz-Zimm type for various numbers of coupled chains in a double logarithmic plot. Note fory=l the Schulz-Zimm distribution becomes the most probable distribution in the limit of/ l the Poisson distribution is eventually obtained. In all cases the weight average degree of polymerization was 100. The narrowing of the distribution with the number of coupled chains is particularly well seen in the double logarithmic presentation...

If the variation were completely unpredictable, there would be no hope of rational planning to take it into account. Usually, however, although it is not possible to predict that a given occurrence will certainly happen, it is possible to assign a probability for any particular occurrence. If this is done for all possible occurrences, then, in effect, a probability distribution function has been defined. Certain types of such distributions can be derived mathematically to fit special situations. The normal, Poisson, and binomial distributions are frequently encountered in practice. 

Determine the probability distribution of in equilibrium, including its normalizing constant. Also find the variance and show that it does not agree with a Poisson distribution. 

The distribution (6.6) is the multivariate generalization of the binomial distribution. Now consider an ensemble of similar systems in which the total N is not constant but distributed according to Poisson with average . Then the probability distribution in this grand ensemble is... 

A process with independent increments can be generated by compounding Poisson processes in the following way. Take a random set of dots on the time axis forming shot noise as in (II.3.14) the density fx will now be called p. Define a process Z(t) by stipulating that, at each dot, Z jumps by an amount z (positive or negative), which is random with probability density w(z). Clearly the increment of Z between t and t + T is independent of previous history and its probability distribution has the form (IV.4.7). It is easy to compute. 

With such a model, the ion-ion interactions are obtained by calculating the most probable distribution of ions around any central ion and then evaluating the energy of the configuration. If (r) is the spherically symmetrical potential in the solution at a distance r from a central ion i of charge z 8, then (r) will be made up of two parts ZxZlDv the coulombic field due to the central ion, and an additional part, ai(r), due to the distribution of the other ions in the solution around t. The potentials at(r) and must satisfy Poisson s equation = — 47rp/D at every point... 

This is a continuous function for the experimental variables, which is used as a convenient mathematical idealisation to describe the distribution of finite numbers of results. The factor 1 /(ay/lji) is a constant such that the total area under the probability distribution curve is unity. The mean value is given by p and the variance by a2. The variance in the Gaussian distribution corresponds to the standard deviation s in Eqn. 8.3. Figure 8-3 illustrates the Gaussian distribution calculated with the same parameters used to obtain the Poisson distribution in Figure 8-2, i.e. a mean of 40 and a standard deviation of V40. It can be seen that the two distributions are similar, and that the Poisson distribution is very dosely approximated by the continuous Gaussian curve. 

To use the MPN method, a probability distribution called the Poisson distribution is used. From any good book on probability, the probability (also called probability function) that a random variable X following the Poisson distribution will have a value y is... 

The second important point on which the CICR technique is based is the strict control of the average number of reactants deposited on the clusters. This is is achieved by using the pick-up technique originally developed by Scoles and coworkers [291]. It consists in capturing the reactants by sticky collisions between the clusters and a low-pressure gas. Of course, the number of particles trapped is not the same for every cluster, but the important point is that the capture process has known statistics, being a random Poisson process. Hence the probability distribution Pq (m ) of finding exactly q reactant molecule per cluster follows the Poisson law of order q ... 

Numerical calculations have been carried out for (i) ternary systems consisting of rods of two lengths x and x and a diluent with x = 1 (ii) solutions of polydisperse rods having a most probable distribution (iii) a Poisson distribution of rods in solution and (iv) various Gaussian distributions of rods in a diluent In all cases longer rods are preferentially partitioned into the nematic phase. For Xa = 2xb in case (i) the ratio of the concentration of either of the species in one of... 

The problem of real time in the algorithmic formulation of dynamic Monte Carlo has been solved by Fichthorn and Weinberg. [34] They replaced the reaction probabilities by rate constants, and assumed that the probability distribution Prx(t) of the time that a reaction occurs is a Poisson process i.e., it is given by... 

Fig. 24.8. Computational simulation analysis of conformational dynamics in T4 lysozyme enzymatic reaction, (a) Histograms of fopen calculated from a simulated single-molecule conformational change trajectory, assuming a multiple consecutive Poisson rate processes representing multiple ramdom walk steps, (b) Two-dimensional joint probability distributions <5 (tj, Tj+i) of adjacent pair fopen times. The distribution <5(ri, Ti+i) shows clearly a characteristic diagonal feature of memory effect in the topen, reflecting that a long topen time tends to be followed by a long one and a short fopen time tends to be followed by a short one...

Assume that a total of N photon counts were registered by the detector. The spread, a, in this case is defined by the Poisson s probability distribution... 

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