By Eliane Regina Rodrigues, Jorge Alberto Achcar

In this short we reflect on a few stochastic types that could be used to review difficulties relating to environmental concerns, particularly, pollution. The effect of publicity to air toxins on people's health and wellbeing is a truly transparent and good documented topic. for this reason, you will need to to procure how you can expect or clarify the behaviour of toxins regularly. reckoning on the kind of query that one is drawn to answering, there are a number of of how learning that challenge. between them we may possibly quote, research of the time sequence of the toxins' measurements, research of the data bought without delay from the information, for example, day-by-day, weekly or per 30 days averages and conventional deviations. differently to review the behaviour of toxins quite often is thru mathematical types. within the mathematical framework we can have for example deterministic or stochastic types. the kind of types that we will ponder during this short are the stochastic ones.

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**Additional info for Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies**

**Example text**

If what we are interested in is estimating the probability of the length of the next inter-surpassing time, given that the present one is of length s, then we have the following. Let Spres and Snext be the most recent and the next interexceedance interval, respectively. Then by the properties of the Poisson process (see for instance [71]), the probability of having Snext greater than t is P(Snext > t | Spres = s) = e−λ t , t ≥ 0. In order to give a numerical example, we have generated a sample of size 1,000 of W53 , and from that we have obtained a sample of λ53 .

2) to calculate the probability that a given environmental threshold is surpassed a number of times within a time interval of interest. Time homogeneous Poisson processes have been used to model several types of problems mainly those of determining the count of occurrences of events. Javits [45] considers that type of process to study the problem of counting the number of surpassings of an environmental threshold by the ozone concentration. 12 ppm. 12 ppm in one year was on average 1 and that of three years was on average equal to 3.

Hence, the difference that might exist between the estimated mean using Model II and the one using Model III is governed mainly by the behavior of the variables Wi , I = 1, 2, . . , N. In Fig. 1 [8] we have the plots of the observed (continuous line) and estimated posterior means for all regions when Models I, II, and III (dashed, dotted, and dashdotted lines, respectively) are used. 2 Homogeneous Poisson Models 35 80 80 Violations-NW 100 Violations-NE 100 60 40 20 60 40 20 0 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Period 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Period 80 80 Violations-SE 100 Violations-CE 100 60 40 20 60 40 20 0 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Period 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Period 100 Violations-SW 80 60 40 20 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Period Fig.