# Exploiting Outage and Error Probability
## Appendix B Derivation of \(f_{\gamma_{d}}\)  
\(\gamma\) follows exponential distribution with mean \(\gamma\). By definition of PDF:  
\[f_{\gamma_{SD}}=\frac{1}{\tilde{\gamma}_{SD}}exp(-\gamma_{SD}/\tilde{\gamma}_ {SD}) \tag{20}\]  
\[f_{\gamma_{R_{i}}}=\frac{1}{\tilde{\gamma}_{R_{i}}}exp(-\gamma_{R_{i}}/\tilde {\gamma}_{R_{i}}),\bigvee\;i\;\epsilon\;\{1,2,3\} \tag{21}\]  
\(f_{\gamma_{d}}\) is calculated by the convolution of above equations. By parts convolution, \(f_{\gamma_{1}}(x)\) is derived as [15]:  
\[ \begin{array}{rcl}f_{\gamma_{1}}(x)&=&\frac{1}{\tilde{\gamma}_{SD}}exp(-x/ \tilde{\gamma}_{SD})*\frac{1}{\tilde{\gamma}_{R_{i}}}exp(-x/\tilde{\gamma}_{R _{1}})\\ &=&\frac{1}{\tilde{\gamma}_{SD}\tilde{\gamma}_{R_{1}}}\int_{0}^{2}exp(\frac{- \tilde{\gamma}_{R_{1}}x^{-}-\tilde{\gamma}_{SD}x^{2}+\tilde{\gamma}_{SD}x}{ \tilde{\gamma}_{SD}\tilde{\gamma}_{R_{1}}})dx\\ &=&\frac{1}{\tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{1}}}exp(-z/\tilde{\gamma} _{SD})-exp(-z/\tilde{\gamma}_{R_{1}})\end{array}  \tag{22}\]  
Similarly, convolution of \(f_{\gamma_{1}}\) and \(f_{\gamma_{R_{2}}}\) is calculated as:\[ \begin{array}{lll}f_{\gamma_{2}}(x)&=&\frac{1}{\tilde{\gamma}_{SD}-\tilde{\gamma}_ {R_{1}}}exp(-x/\gamma_{SD})-exp(-x/\tilde{\gamma}_{R_{1}})*\frac{1}{\tilde{\gamma }_{R_{2}}}exp(-x/\tilde{\gamma}_{R_{2}})\\ &=&\frac{1}{\tilde{\gamma}_{R_{2}}(\tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{1}}) }\int_{0}^{z}\left[\left[exp(\frac{-\tilde{\gamma}_{R_{2}}x-\gamma_{SD}z+ \tilde{\gamma}_{SD}x}{\tilde{\gamma}_{R_{2}}\tilde{\gamma}_{SD}})\right]- \left[exp(\frac{-\tilde{\gamma}_{R_{2}}x-\tilde{\gamma}_{R_{1}}z+\tilde{\gamma }_{R_{1}}x}{\tilde{\gamma}_{R_{1}}\tilde{\gamma}_{R_{2}}})\right]\right]dx\\ &=&\frac{\left[\frac{\tilde{\gamma}_{SD}}{\tilde{\gamma}_{SD}-\tilde{\gamma}_ {R_{2}}}[exp(-z/\gamma_{SD})-exp(-z/\tilde{\gamma}_{R_{2}})]-\frac{\tilde{ \gamma}_{R_{1}}}{\tilde{\gamma}_{R_{1}}-\tilde{\gamma}_{R_{2}}}[exp(-z/\tilde {\gamma}_{R_{1}})-exp(-z/\tilde{\gamma}_{R_{2}})]\right]}{(\tilde{\gamma}_{ SD}-\tilde{\gamma}_{R_{1}})}\end{array}  \tag{12}\]  
Third convolution of \(f_{\gamma_{2}}\) with \(f_{\gamma_{R_{3}}}\) is calculated as:  
\[ \begin{array}{lll}f_{\gamma_{3}}(x)&=&\frac{\left[\frac{\tilde{\gamma}_{SD} }{\tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{2}}}[exp(-x/\gamma_{SD})-exp(-x/\tilde {\gamma}_{R_{2}})]-\frac{\tilde{\gamma}_{R_{1}}}{\tilde{\gamma}_{R_{1}}-\tilde {\gamma}_{R_{2}}}[exp(-x/\tilde{\gamma}_{R_{1}})-exp(-x/\tilde{\gamma}_{R_{2}}) ]\right]}{(\tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{1}})}\\ &=&\frac{\gamma_{SD}}{\tilde{\gamma}_{R_{3}}(\tilde{\gamma}_{SD}-\tilde{ \gamma}_{R_{2}})}\int_{0}^{z}\frac{\left[exp(\frac{-\tilde{\gamma}_{R_{3}}x- \tilde{\gamma}_{SD}z+\tilde{\gamma}_{SD}x}{\tilde{\gamma}_{SD}\tilde{\gamma}_ {R_{3}}})-exp(\frac{-\tilde{\gamma}_{R_{3}}x-\tilde{\gamma}_{R_{2}}z+\tilde{ \gamma}_{R_{2}}x}{\tilde{\gamma}_{R_{2}}\tilde{\gamma}_{R_{3}}})\right]dx}{ \tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{1}}}\\ &&-\frac{\tilde{\gamma}_{R_{1}}}{\tilde{\gamma}_{R_{3}}(\tilde{\gamma}_{R_{1}}- \tilde{\gamma}_{R_{2}})}\int_{0}^{z}\frac{\left[exp(\frac{-\tilde{\gamma}_{R_{3 }}x-\tilde{\gamma}_{R_{1}}z+\tilde{\gamma}_{R_{1}}x}{\tilde{\gamma}_{R_{1}} \tilde{\gamma}_{R_{3}}})-exp(\frac{-\tilde{\gamma}_{R_{3}}x-\tilde{\gamma}_{R_ {2}}z+\tilde{\gamma}_{R_{2}}x}{\tilde{\gamma}_{R_{2}}\tilde{\gamma}_{R_{3}}}) \right]dx}{\tilde{\gamma}_{SD}-\tilde{\gamma}_{R_{1}}}\end{array}  \tag{13}\]  
Integrating and doing some necessary manipulations, PDF of \(f_{\gamma_{d}}\) is given by:  
\[