## 2 Statistical Characteristics of \(\alpha\)-\(\mu\)/IGA Distribution
### Univariate Case
By letting \(t=(n-1)/z\) in (5) and performing some straightforward manipulations and several the changes of variables, (7) can be approximated as  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]0  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]1 PDFs of the \(\alpha\)-\(\mu\)/IGA and \(\alpha\)-\(\mathcal{F}\) models for different channel parameters with \(\overline{\gamma}=1\). where \(\xi_{i}=\omega_{i}t_{i}^{\mu-1}/\Gamma(n)\), \(\psi_{i}=\overline{\gamma}(n-1)/(\beta t_{i})\), \(N\) denotes the number of truncation terms, and \(\omega_{i}\) and \(t_{i}\) are the abscissas and weight factors for the Gaussian-Laguerre integration, respectively. Here, we name the novel form in (10) as a mixed \(\alpha\)-\(\mu\) distribution model. In particular, this model is a generalized version of the MG distribution and can be referred to as the generalized MG (GMG) distribution. Most noteworthy is that the \(\alpha\)-X multipath and \(\alpha\)-\(\mu\)-Y composited fading distributions can be simplified as this mixture sum form, where X denotes \(\eta\)-\(\mu\) or \(\kappa\)-\(\mu\), and Y denotes LN, Gamma, or \(\alpha\)-\(\mu\)[11] distribution models. Based on the definition in (8) and (9), and using (10), the corresponding approximated forms of the CDF and the MGF of \(\gamma\) can be obtained, respectively, as  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]2  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]3  
where \(\gamma(\cdot,\cdot)\) is the lower incomplete gamma function defined in [37] (Equation (8.310.1)). In order to prove the accuracy of the approximated distribution models, two classical criteria of error are often adopted to evaluate the error between the exact PDF and the approximated PDF in [41], namely the mean square error (MSE) and the Kullback-Leibler divergence (KLD). For the purposes of comparison, the approximated PDF of \(\gamma\) based on the mixed IGA model can be expressed as [26]  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]4  
where \(\xi_{i}=\omega_{i}t_{i}^{\mu-1}/\Gamma(\mu)\), \(\eta_{i}=\overline{\gamma}(n-1)(t_{i}/\beta)^{2/\alpha}\). In Figure 2a-d, we compare the exact PDF in (7), the mixed \(\alpha\)-\(\mu\) PDF in (10), and the mixed IGA PDF in (13) under different conditions, and present the impacts of various parameters on the PDFs. At the same time, the Monte Carlo simulations are also provided to validate the accuracy of the approximated distributions. From Figure 2a-d, it can be observed that the PDF curves of the mixed IGA model show the larger divergences when the values of the truncated terms (\(N\)), \(\alpha\), and/or \(\mu\) are smaller (for example, \(N=8\), \(12\) in Figure 2a, \(\alpha=1.5\) in Figure 2b, and \(\mu=1.5\) in Figure 2d), or the values of the shadowing parameter, \(n\), become larger (for example, \(n=20\) in Figure 2c), whereas it is interesting that the PDF curves of the mixed \(\alpha\)-\(\mu\) achieve good agreements with the exact PDF in (7) in most of the cases except the case where \(\alpha=5\) and \(N=8\) in Figure 2b. Although increasing the value of \(N\) can reduce the divergences between the exact PDF and the mixed IGA, it renders more computation complexity and a longer running time. Furthermore, with the aid of the definitions of MSE and KLD in [42], the error analytical results are calculated under various fading scenarios in \[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]5 From Table 1, we can also obtain the same conclusions as ones from Figure 2a-d. Despite the mixed \(\alpha\)-\(\mu\) distribution being used to simplify the \(\alpha\)-\(\kappa\)-\(\mu\) shadowed model in [43], its fading parameters (\(\mu\) and \(m\)) are constrained to be positive integers and cannot be extended to other models, as well as having no application for it. To the best of the authors' knowledge, the mixed \(\alpha\)-\(\mu\) PDF in (10) is novel and has not yet been reported in the open technical papers.