Poisson Probability Example

Suppose accident data revealed that 4 defective components were found in 30,000 units shipped to the marketplace. A product liability prevention program was estab- 

The first step is to find the expected number of failures, M. This is determined by multiplying the number of exposure units (N) by the probability of one failure (p). There were 50,000 units shipped, which represents the exposure. The probability of a defective unit is 4/30,000= 1.33 10 based upon the previous units shipped. With the 50,000 units, one would expect 50,000 1.33 10 = 6.7 defective units. 

The next step is to plug the values into the formula. In the Poisson probability equation, the term e is the base of a natural logarithm, which is approximately equal to 2.718, and the variable X represents the specific number of defective units being investigated. In this example, X = 3. Therefore, we have  

If one uses the cutoff point of. 05 or 5% random chance, then the result of a 6% chance of finding 3 defective units in 50,000 is not significant therefore, the safety manager can conclude the product liability program has not significantly reduced the number of defective units being shipped. 

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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,... 

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 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... 

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. 

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... 

Similar mathematical solution can be derived from a Poisson distribution of random events in 2D space. The probability that 2D separation space will be covered by peaks in ideally orthogonal separation is analogical to an example where balls are randomly thrown in 2D space divided into uniform bins. The general relationship between the number of events K (number of balls, peaks, etc.) and the number of bins occupied F (bins containing one or more balls, peaks, etc.) is described by Equation 12.3, where N is the number of available bins (peak capacity in 2DLC). 

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. 

The limiting probabilities p for the number of ionic species in the system can be established in a manner similar to the production/decomposition of a radioactive particle possessing the same probabilistic properties6 (Example 38.3, pp. 237-238). The result of an involved algebraic manipulation, itself a Poisson distribution ... 

An important example of a one-step process with constant transition probabilities is the Poisson process, defined by... 

The Poisson distribution can be applied also to describe the action of detectors. For example, suppose the interaction of a 7-ray photon with an inefficient scintillator produced, on average, 3.3 photoelectrons from the photocathode. The probability of producing no photoelectrons (not seeing the event) is given by the Poisson distribution as ... 

The Poisson distribution has been used several times in this text in connection with problems involving small independent probabilities. For example, suppose we desire to find the distribution of v random particles on an 7V-square uniform grid, N being large and each square equally accessible the distribution will then be given by the expression... 

Equation (5) is an example of the Poisson process in probability theory [9]. S decays slowly with t, so that enough sites survive even at long times to continue forming new coke. This simple model lends itself naturally to the distinction of coke depositing on clean sites, monolayer coke concentration C , and coke depositing on already-coked sites, multilayer coke concentration Cm- That is, C (/) = CJf) + CM t) at all times. A simple site balance yields... 

In the example, we choose x = 0 and p 0 and consider only the broadening of the momentum distribution V (p) by varying its width a. The results can be seen in Fig. 2 The transition probability for a = 0.1 is shown in Fig. 2 (a). The distribution is centered around n - 5 and has the Poisson-like shape of a Franck-Condon type of interaction, as is expected for a quasiconstant interaction with small a. With increasing a, the maximum is shifted slightly towards higher values of n. But the most obvious effect is the broadening of the probability distribution. For a — 00, an extremely narrow position wave packet with its center at x = 0 is excited on the upper potential. In this limit, the probability distribution is... 

Following are some examples of frequently encountered probability distributions Poisson distribution. This is the discrete distribution... 

The Poisson distribution describes the probability of a discrete number of events occurring within a fixed interval, given that the probability of the event occurring is independent of the size of the interval. For example, suppose that a manufacturing defect occurs with an average rate of occurrence p and the products are manufactured over an interval n. The expected number of defects is clearly np. The Poisson distribution is one way of describing the probability that x defects will occur in an interval n. This distribution is a generalization of the binomial distribution to an infinite number of trials. The mathematical form of the Poisson distribution [2] is... 

For example, it is evident that antibiotic molecules are distributed randomly among ribosomes if the ribosomes are not clumped. Such an event would be expected to follow the Poisson distribution Pn (m) = m"e " /n, in which P is the probability that any antibiotic molecule ( ) at a binding site of the ribosome will be detected, and m is the mean number of drug molecules bound per ribosome. When the ratio of erythromycin molecules to ribosomal particles is one to one, it follows from the Poisson series that the proportion of ribosomes ... 

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