Novel simple forms for the probability density function (PDF) and the cumulative distribution function (CDF) of two correlated Rician random variables (RVs) are derived. The new representations for the bivariate Rician PDF and CDF are given in the form of single integral solutions in terms of well known functions that are available in common mathematical software packages. The general case of non-identically distributed Rician RVs is considered in this analysis. The new form of the bivariate Rician cumulative distribution function is applied to give a new expression for the outage probability of dual selection combining diversity operating in correlated Rician fading.