## 3 Analysis of Success Probability
### Conditional Success Probability
#### 3.2.1 Conditional Success Probability in Primary Network  
The conditional success probability at any typical PR \(y_{k}^{p}\) (expressed as \(\mu_{y_{k}^{p}}^{!}\)) is thus given as  
\[\mu_{y_{k}^{p}}^{!}=P(SINR_{y_{k}^{p}}>\theta_{p}|\Phi,\forall\Phi=\Phi_{p}^{ I}\cup\Phi_{s}^{I}). \tag{16}\]  
**Lemma 1**.: _In the conventional underlay mode, the conditional success probability \(\mu_{y_{k}^{p}}^{!}\) can be expressed as_  
\[\mu_{y_{k}^{p}}^{!}=\exp(\frac{\theta_{p}r_{p}^{p}\sigma^{2}}{P_{p}})\prod_{x_ {j}^{s}\in\Phi_{j}^{s}\cup x_{k}^{p}}\Big{(}\frac{v_{p}}{1+\theta_{p}r_{p}^{p} ||x_{j}^{p}||^{-\alpha}}+1-v_{p}\Big{)}\prod_{x_{j}^{s}\in\Phi_{j}^{s}}\big{(} \frac{v_{s}}{1+\theta_{p}r_{p}^{p}||x_{j}^{s}||^{-\alpha}\frac{\theta_{p}^{ MI}}{P_{s}^{s}}}+1-v_{s}\Big{)}. \tag{17}\]Proof.: By substituting (2) into (16), the analysis for the success probability can be obtained conditioned on \(\Phi\). The proof is straightforward from Appendix A of [20].  
However, the analysis in (17) is difficult to solve, hence its b-th moment as \(\sigma^{2}\to 0\) is obtained as  
\[M_{b}=E_{y_{k}^{p}}[(\mu_{y_{k}^{p}}^{uU})^{b}]=E_{r_{p}}\Big{[}E\prod_{x_{j}^{ p}\in\Phi_{p}^{1}\setminus x_{k}^{p}}\Big{(}1-\frac{E[v_{p}|\Phi]}{1+\frac{||x_{j}^{ p}||^{\alpha}}{\theta_{p}r_{p}^{\alpha}}}\Big{)}^{b}E\prod_{x_{j}^{p}\in\Phi_{p}^{1} }\Big{(}1-\frac{E[v_{s}|\Phi]}{1+\frac{||x_{j}^{p}||^{\alpha}}{\theta_{p}r_{p} ^{\alpha}}\frac{P_{p}}{P_{p}^{1}}}\Big{)}^{b}\Big{]}. \tag{18}\]  
Following the analysis in [28; 30], the b-th moment of \(\mu_{y_{k}^{p}}^{uU}\) can be expressed as  
\[M_{b}=\exp\Big{\{}-\lambda_{p}\theta_{p}^{\delta}r_{p}^{2}\frac{\pi^{2}\delta }{\sin(\pi\delta)}\sum_{k=1}^{\infty}\binom{b}{k}\binom{\delta-1}{k-1}v_{p}^{k }\Big{\}}\exp\Big{\{}-\lambda_{s}\Big{(}\frac{\theta_{p}P_{s}^{U}}{P_{p}} \Big{)}^{\delta}r_{p}^{2}\frac{\pi^{2}\delta}{\sin(\pi\delta)}\sum_{k=1}^{ \infty}\binom{b}{k}\binom{\delta-1}{k-1}v_{s}^{k}\Big{\}}. \tag{19}\]  
The meta distribution can be approximated following the approach presented in [30; 31] with the beta distribution as  
\[P[\mu_{y_{k}^{p}}^{uU}\leq x]=I_{x}(\beta_{1},\beta_{2}),\]  
where the parameter \(I_{x}(.,.)\) is known as the regularized incomplete beta function and  
\[\beta_{1}=\frac{M_{1}(M_{1}-M_{2})}{M_{2}-M_{1}^{2}},\]  
\[\beta_{2}=\frac{(M_{1}-M_{2})(1-M_{1})}{M_{2}-M_{1}^{2}}.\]  
\(M_{1}\) and \(M_{2}\) are the first and second moment of \(\mu_{y_{k}^{p}}^{uU}\) given in (19).  
**Lemma 2**.: _The conditional success probability at any typical PR \(y_{k}^{p}\) in the conventional overlay mode is given as_  
\[\mu_{y_{k}^{p}}^{\mathrm{IO}}=\exp(\frac{\theta_{p}r_{p}^{\alpha}\sigma^{2}}{ P_{p}})\prod_{x_{j}^{p}\in\Phi_{p}^{1}\setminus x_{k}^{p}}\Big{(}\frac{v_{p}}{1+ \theta_{p}r_{p}^{\alpha}||x_{j}^{p}||^{-\alpha}}+1-v_{p}\Big{)}.\]  
Its b-th moment as \(\sigma^{2}\to 0\) is obtained as  
\[M_{b,overlap}=\exp\Big{\{}-\lambda_{p}\theta_{p}^{\delta}r_{p}^{2}\frac{\pi^{ 2}\delta}{\sin(\pi\delta)}\sum_{k=1}^{\infty}\binom{b}{k}\binom{\delta-1}{k-1 }v_{p}^{k}\Big{\}}. \tag{20}\]  
Proof.: Under the conventional overlay CRN, the interference from STs is neglected. Hence, the conditional success probability depends only on the interference from other active PTs as in Section 3.1. Finally, the case of misdetection is straightforward from (19) by taking \(P_{s}=P_{s}^{O}\), while the analysis for the hybrid case is straightforward from (4).