## 4 Analysis
### Probabilistic Model (PM) for the Distributed Approach  
This subsection introduces a particular problem considering the mean and variance of the overlap of geometrical figures [31]. Given a number of circles placed at random on a plane so that each one may have some or all of its area inside a target area, with reference to the bombing studies, [32] uses the  
Figure 5: The three-layer perceptron neural network architecture for analyzing the CASA scheme. concept of geometry and of probability to explore the fundamentals of this type of problem. The result is represented as the following theorem. **Theorem 1**: _Let \(X\) be a random Lebesgue measurable subset of n-dimensional Euclidean space \(E_{n}\), with measure \(\mu(X)\). For any point \(x\) of \(E_{n}\), let \(p(x)=Pr(x\in X)\). Assuming that the function \(g(x,X)\) is a measurable function of the pair \((x,X)\) with \(g(x,X)=1\) for \(x\in X\) and zero elsewhere, the expected value of the measure X is the Lebesgue integral of the function \(p(x)\) over \(E_{n}\)._  
Suppose that \(A\) and \(C\) are the interior of the closed curves. Let the subset \(X\) be the part of a region \(A\) in \(E_{2}\) which is covered by _z C_'s dropped independently and randomly. Denote a reference point \(Q\) as the centra of the area \(C\) and assume that there is a frequency distribution \(\phi(x,y)\) of the position \((x,y)\) of \(Q\). Based on the above assumptions, now we consider the moments of the area \(Y=A-X\) (i.e. the area of \(A\) not covered by the _z C_'s). Referring to Theorem 1, the probability of a point \((x_{1},y_{1})\) in \(Y\) not being covered by a \(C\) is:  
\[q(x_{1},y_{1})=\int_{T-\overline{C}(x_{1},y_{1})}\int\phi(x,y)dxdy, \tag{14}\]  
where \(T-\overline{C}(x_{1},y_{1})\) is the part of \(T\) exterior to the area \(C\). Therefore, the first moment of \(Y\), in the case of _z C_'s, yields:  
\[\mu_{Y}^{(1)}=\int_{A}\int q^{z}(x_{1},y_{1})dx_{1}dy_{1}. \tag{15}\]  
similarly, for the _m_th moment, the probability that the points \((x_{1},y_{1})\), \((x_{2},y_{2})\), \(\ldots\), \((x_{m},y_{m})\) are not covered by a \(C\) is:  
\[q(x_{1},y_{1},x_{2},y_{2},\ldots,x_{m},y_{m})=\int_{T-\overline{C}_{1}- \overline{C}_{2}-\ldots-\overline{C}_{m}}\int\phi(x,y)dxdy, \tag{16}\]  
where \(T-\overline{C}_{1}-\overline{C}_{2}-\ldots-\overline{C}_{m}\) is the area of \(T\) outside _C_'s centered at \((x_{1},y_{1})\), \((x_{2},y_{2})\), \(\ldots\), \((x_{m},y_{m})\). Thus, the _m_th moment in the case of _z C_'s is given by:  
\[\mu_{Y}^{(m)}=\int_{A}\int\cdots\int_{A}\int q^{z}(x_{1},y_{1},x_{2},y_{2}, \ldots,x_{m},y_{m})dx_{1}dy_{1}dx_{2}dy_{2}\ldots dx_{m}dy_{m}. \tag{17}\]  
Accordingly, in our case we may interpret \(T\) as the sensing field in a cluster, let \(A\) be the sensing field of a given sensor, let \(X\) be the area covered by its neighboring cluster members, and \(Y\) will represent the sensing area of a given sensor not covered by its neighboring cluster members. Denote the parameters \(z\) and \(m\) as the number of unconnected cluster members and the neighboring cluster members, respectively. Therefore, the _m_th moment \(\mu_{Y}^{(m)}\) may describe the fraction of \(A\) of a given sensor not covered by its \(m\) neighboring cluster members. Lindeberg Theorem  
This subsection reviews the probability that is used when analyzing the performance of the model. Readers may refer [29] for a complete discussion and proof of the theorem. Suppose for each \(n\)  
\[(X_{11},X_{12},\ldots,X_{1r_{1}})\]\[ \begin{array}{c}(X_{21},X_{22},\ldots,X_{2r_{2}})\\ \vdots\\ (X_{n1},X_{n2},\ldots,X_{nr_{n}})\end{array}  \tag{18}\]  
are independent random vectors. The probability space may change with \(n\) and (18) is called a _Triangular Array_ of random variables. Put \(S_{n}=X_{n1}+\cdots+X_{nr_{n}}\). In the network application, let \(X_{ni}\) be \(X_{i}\) and let \(X_{i}\) take the values 1 and 0 with probability \(p_{i}\) and \(q_{i}=1-p_{i}\).