Abstract

In Holm and Alouini (2004 Holm, H., and M. Alouini. 2004. Sum and difference of two squared correlated Nakagami variates in connection with the Mckay distribution. IEEE Transactions on Communications 52 (8):1367–76. doi:10.1109/TCOMM.2004.833019.[Crossref], [Web of Science ®] , [Google Scholar]), an explicit formula for the probability density function of ${\lambda }_1^{-1}{Z}_r+{\lambda }_2^{-1}{\tilde{Z}}_r$ was obtained, where ${\lambda }_1\ne {\lambda }_2\ne 0,Z_r,{\tilde{Z}}_r$ are independent identical random variables with $\Gamma (r,1)$ distribution. In this paper, we use a recent technique from Gaunt (2018 Gaunt, R. E. 2018. Products of normal, beta and gamma random variables: Stein operators and distributional theory. Brazilian Journal of Probability and Statistics 32 (2):437–66.[Crossref], [Web of Science ®] , [Google Scholar]) involving Stein’s method literature to obtain some simple and new proofs, which could also be used in other problems as a general method.

摘要

在 Holm 和 Alouini ( 2004 Holm, H.和M. Alouini。2004 年。相关 Nakagami 变量的两个平方的和与差与 Mckay 分布有关。IEEE 通讯汇刊52 (8): 1367 – 76。内政部：10.1109/TCOMM.2004.833019。[Crossref], [Web of Science®]  , [Google Scholar] ), 概率密度函数的显式公式${\lambda }_{\mathrm{1个}}^{-\mathrm{1个}}{Z}_r+{\lambda }_{\mathrm{2个}}^{-\mathrm{1个}}{\tilde{Z}}_r$获得，其中${\lambda }_{\mathrm{1个}}\ne {\lambda }_{\mathrm{2个}}\ne 0,Z_r,{\tilde{Z}}_r$是独立相同的随机变量$\Gamma (r,\mathrm{1个})$分配。在本文中，我们使用了 Gaunt ( 2018 憔悴，重新 2018 年。正态、β 和伽马随机变量的乘积：Stein 算子和分布理论。巴西概率统计杂志32 (2): 437 – 66。[Crossref], [Web of Science®]  , [Google Scholar] )涉及Stein的方法文献，得到一些简单的新证明，也可以作为通用方法用于其他问题。