## 4 Outage Performance  
Outage probability reflects the ability of the user in the system to receive and decode data correctly. The outage probabilities of U\({}_{1}\) and U\({}_{2}\) are obtained based on the cumulative distribution functions (CDFs) and probability density function (PDF). A characteristics function is defined, i.e., \(I_{\{A\}}=1\) for the event \(A\) occurs and \(I_{\{A\}}=0\) for otherwise, which will be used in the following. In order to facilitate the derivation of outage probability expressions, the following propositions are presented. **Proposition 1**. : _Defining a random variable \(\rho|h|^{2}\) where \(\rho=P/\sigma^{2}\), \(P=min\Big{\{}I_{th}/|h_{0}|^{2},P_{0}\Big{\}}\), \(h_{0}\sim CN(0,\lambda_{0})\), \(h\sim CN(0,\lambda)\) and \(\sigma^{2}\), \(I_{th}\), \(P_{0}\) are constants, the CDF of \(\rho|h|^{2}\) is given by_  
\[F_{\rho|h|^{2}}(x)=1-exp\bigg{(}-\frac{\sigma^{2}x}{\lambda P_{0}}\bigg{)} \bigg{(}1-\frac{\lambda_{0}\sigma^{2}x}{\lambda I_{th}+\lambda_{0}\sigma^{2}x }exp\bigg{(}-\frac{I_{th}}{\lambda_{0}P_{0}}\bigg{)}\bigg{)}. \tag{22}\]  
Proof. : Noting that \(F_{|h_{0}|^{2}}(x)=1-exp(-x/\lambda_{0})\), since \(h_{0}\sim CN(0,\lambda_{0})\), the random variable \(P=min\Big{\{}I_{th}/|h_{0}|^{2},P_{0}\Big{\}}\) can be written as  
\[P=\left\{ \begin{array}{cc}P_{0},&\left|h_{0}\right|^{2}\leq I_{th}/P_{0},\\ I_{th}/\left|h_{0}\right|^{2},&\left|h_{0}\right|^{2}\geq I_{th}/P_{0}.\end{array} \right. \tag{23}\]  
Therefore, for a given \(\left|h_{0}\right|^{2}\), the CDF of the random variable \(\rho|h|^{2}\) can be given by  
\[F_{\rho|h|^{2}}(x)=\int_{0}^{\infty}Pr\Big{\{}\rho|h|^{2}\leq x\left\|h_{0} \right|^{2}=y\Big{\}}f_{|h_{0}|^{2}}(y)dy \tag{24}\]  
\[\text{where }Pr\Big{\{}\rho|h|^{2}\leq x\left\|h_{0}\right|^{2}=y\Big{\}}\]  
\[=\left(1-exp\Big{(}-\frac{\sigma^{2}x}{\lambda P_{0}}\Big{)}\right)I_{\{y\leq \frac{I_{th}}{P_{0}}\}}+\left(1-exp\Big{(}-\frac{\sigma^{2}xy}{\lambda I_{th}} \Big{)}\right)I_{\{y>\frac{I_{th}}{P_{0}}\}}; \tag{25}\]  
\[f_{|h_{0}|^{2}}(y)=1/\lambda_{0}exp(-y/\lambda_{0}) \tag{26}\]  
Substituting Equations (25) and (26) into (24), and after some algebraic manipulations, we can obtain Equation (22), which completes the proof for Proposition 1. \(\square\)  
**Proposition 2**. : _Defining a random variable \(\rho^{\prime}|h|^{2}+1\) where \(\rho^{\prime}=P^{\prime}/\sigma^{2}\), \(h\sim CN(0,\lambda)\)\(h\sim CN(0,\lambda)\) and \(\sigma^{2}\), \(P^{\prime}\) are constants, the CDF and PDF of \(\rho^{\prime}|h|^{2}+1\) are given, respectively, as_  
\[F_{\rho^{\prime}|h|^{2}+1}(x)=\bigg{[}1-exp\bigg{(}-\frac{\sigma^{2}(x-1)}{ \lambda P^{\prime}}\bigg{)}\bigg{]}I_{\{x\geq 1\}}\text{ and }f_{\rho^{\prime}|h|^{2}+1}(x)=\bigg{[}\frac{\sigma^{2}}{\lambda P }exp\bigg{(}-\frac{\sigma^{2}(x-1)}{\lambda P^{\prime}}\bigg{)}\bigg{]}I_{\{x \geq 1\}} \tag{27}\]  
Proof obviously. The following two theorems provide the closed-form expressions for the outage probabilities achieved by the two-stage adaptive relay selection and power allocation of two users in the proposed cooperative underlay CR-NOMA network. **Theorem 1. **_The outage probability for \(U_{2}\) in cooperative underlay CR-NOMA networks with the proposed two-stage adaptive relay selection and power allocation strategy is given by_  
\[