Copulas of bivariate Rayleigh and log-normal distributions

The intrinsic relations between multivariate distributions and their marginal distributions can be clearly characterised by copulas. The bivariate Rayleigh and bivariate log-normal distributions play important roles in both signal processing and digital communications. In this reported work, the formula of bivariate Rayleigh copula is improved and the formula of bivariate log-normal copula is derived. These formulas are in terms of the Marcum Q-function or the Gaussian Q-function, both being supported by Matlab. Thus the copula evaluation process can be expedited both analytically and numerically.