# An Accurate Probabilistic Model for TVWS Identification
## 4 Proposed Probabilistic Model
### Working with Normal Distributions
First, for calculating \(P_{fa}\), the left intersection point \"C\" of the pdf of \(W(p)\) and \(X(p)\)  
Figure 7: The curve \(Xo(p)\) of the multi modal distribution of **(a)** can be assumed to follow the normal distribution curve \(X(p)\) of **(b)**, and this assumption summarized in **(c)** allows us to reduce the calculation complexity of the proposed model. Readings taken in three different points. The Probability Density Function (pdf) for the noise \(W(p)\) is plotted in blue, multi modal composite signal \(Xo(p)\) in magenta, and composite signal \(X(p)\) in red. In **(d)**, the pdf of the noise for each location are shown. In **(e)**, the intersection point “\(A\)” of \(W(p)\cap Xo(p)\) and its projection to the power axis “\(A\)” are represented. In **(f)**, the intersection point “\(B\)” of \(W(p)\cap X(p)\), and its projection to the power axis “\(B\)” are represented. will not be considered here, because it is located below the mean value of the noise signal \(\overline{\mathrm{W}}(p)\). And second, for calculating of \(P_{d}\), the right intersection point \"\(B\)\" of the pdf of \(W(p)\) and \(X(p)\), and the right tail of the signal will be considered. Once these assumptions are adopted, then the pdf that represents the noise \(W(p)\) and the composite signal plus noise \(X(p)\) are those ones illustrated in Figure 8.  
The frequently used statistical parameters regarding the sensed signals are \(P_{fa}\), \(P_{d}\), and the missed detection probability (\(P_{m}\)). With the collected information of \(W(p)\) and \(X(p)\), and considering that both signals correspond to normal distributions, other statistical parameters can be calculated, such as mean values of the signals (\(\overline{\mathrm{W}}(p)\), \(\overline{\mathrm{X}}(p)\), or their standard deviation (\(\sigma_{\mathrm{W}}\) and \(\sigma_{\mathrm{X}}\)). An interesting manner to relate the obtained parameters is by using the pdf of the signals, which provides a visual explanation of the proposed method, and it is explained next. To plot the pdf of a signal that follows a normal distribution, it is necessary to apply  
\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}exp\left\{\frac{-1}{2}\left(\frac{x-\overline{ x}(p)}{\sigma}\right)^{2}\right\};\;-\infty<x<\infty \tag{5}\]  
where \(\sigma\) is the standard deviation and \(\overline{\mathrm{x}}(p)\) is the mean of the distribution. By taking the samples of \(X(p)\) and \(W(p)\), the equations that define both pdf curves, are \(f(x_{\mathrm{W}})\) and \(f(x_{X})\) with \(\overline{\mathrm{W}}(p)\) and \(\overline{\mathrm{X}}(p)\) as the means, \(\sigma_{\mathrm{W}}\) and \(\sigma_{\mathrm{X}}\) as the standard deviations respectively, and then the equations are:  
\[f(x_{W})=\frac{1}{\sigma_{\mathrm{W}}\sqrt{2\pi}}exp\left\{\frac{-1}{2}\left( \frac{x-\overline{\mathrm{W}}(p)}{\sigma_{\mathrm{W}}}\right)^{2}\right\} \tag{6}\]  
\[f(x_{X})=\frac{1}{\sigma_{\mathrm{X}}\sqrt{2\pi}}exp\left\{\frac{-1}{2}\left( \frac{x-\overline{\mathrm{X}}(p)}{\sigma_{\mathrm{X}}}\right)^{2}\right\} \tag{7}\]  
Now, considering Equation (2) that shows the composite signal consisting of noise and PU's energy, it is important to compare both pdf.