Poisson Distribution — Cumulative Probabilities

Table 6. Poisson Distribution—Cumulative Probabilities P (continued)...

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,... 

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. 

B) Comparison between calculated and experimental distribution of a water-soluble marker (ferritin) inside POPC vesicles. Detailed data analysis shows that in some cases ferritin can be entrapped with efficiency higher than what expected on theoretical basis (Poisson distribution). Data taken from Berciaz et al. (C) Probability of co-entrapment of all macromolecular components of transcription-translation kit inside lipid vesicles of a given radius. The entrapment of each molecule is modelled as a poissonian process, and the cumulative probability is calculated as product of probabilities of independent events. The curve (a) indicates the probability of entrapping at least one copy of each molecular specie inside the same vesicle. The curve (b) indicates the probability of entrapping at least one copy of each molecular species under the hypothesis that their concentrations are all 50 times higher than the nominal (bulk) concentrations. Adapted from Souza et aP ... 

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. 

The probability of failure would be calculated by multiplying the area of interest s impact surface area, fluence, geometric factor, and attitude factor for each mission phase to give the number of impacts in a certain area of the spaceship. A factor of 2 would typically be applied to any specified fluence data used for calculations to account for uncertainty. A Poisson distribution is then applied to the multiplied value to obtain the probability of failure. Using cumulative probability formulation, the overall probability of micrometeoroid induced failure for the reactor module could be calculated using the results from the individual area of interest calculations. 

---