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DOI: 10.4018/978-1-5225-3270-5.ch005

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TopBinomial distribution (check Box 1) analyses discrete random variables. It is appropriate when an event has only two possible outcomes: Success (p probability of first outcome) or failure (q=1-p, i.e., the probability of second outcome). Recalling that probabilities of all outcomes result in a sum of one, it is mandatory that p+q=1 (p represents the probability of a car being robbed in a particular metropolis, for example, while q equals the complementary probability of it not being stolen). Other appealing features for GIS of this peculiar distribution are: (1) All trials are independent events; (2) All trials are subject to the same probability distribution; (3) The probable sum of all possible combinations (n=1,2,3...N) for a given success probability of p with a size population of N equals one; (4) The N number of trials is considered fixed.

Returning to our car example, this means that the number of the sampled vehicles for that metropolis must be known. Obviously, when the number of trials changes, the number of all possible outcomes and their distributions across the values of the random variable will change too (if we survey six cars instead of four, we will have 64=2^{6} possible outcomes instead of 16=2^{4}). Furthermore, the considered n number of successes (stolen cars) must be a positive integer below or equal to N (see the equation in Box 1).

Box 1. Statistically, the binomial distribution gives the discrete probability distribution, P_{p}(n|N), of obtaining exactly n successes out of N Bernoulli trials

What is the probability of having two or more days of rain in Montevideo, Uruguay, for one single week if the chance of rain is 5% on each day?

Using this example of myGeoffice© output, the answer becomes P(x=2)+P(x=3)+P(x=4) +P(x=5)+P(x=6)+P(x=7)=4.4%. It is also possible to say that the probability of it not raining during the whole week is 69.83%

Brites-Neto and Roncato Duarte (2015) tried to develop maps to identify environmental suitability for the two core species of scorpions of medical importance in São Paulo, Brazil, and to develop spatial configuration parameters for epidemiological surveillance of these venomous animals. By using 54 georeferenced points for Serrulatus, 86 points for Bahiensis and 8 environmental indicators, both authors generated a species distribution model (ESRI^{®} ArcGIS 10.2.2 for Desktop) using omission error and binomial probability for evaluation purposes. Under the emergency warning actions for prevention and surveillance from scorpion stings, the resulting map showed a high environmental suitability of the north, central and southeastern parts of this Brazilian state, confirming the increasing spread of these species. Moreover, the environmental variables that most greatly affected the scorpion distribution model were rain precipitation (28.9%) and tree cover (28.2%) for the Serrulatus; and temperature (45.8%) and thermal amplitude (12.6%) for the Bahiensis.

As a reference (not included in myGeoffice^{©}), the negative binomial distribution concerns to the probability of k successes in a sequence of g independent trials (if we define 1 as tails and 0 as heads, for instance, and we flip a coin repeatedly until the sixth time 1 appears then the probability distribution of the number of 0s that had appeared will be a negative binomial).

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