Many statistical methods say testing correlation and equality of variances, have been developed under the assumptions of independence, identicality and normality of observations. In this talk, we will address how far we can relax the three strong assumptions with special reference to distributions that shares many intrinsic properties of normal distribution but distinct in many respects. These are bivariate t-distribution, bivariate contaminated normal distribution and many other distributions belonging to bivariate elliptical distributions.