In this paper, a generalized form of the multivariate normal distribution has been adopted for investigation. This generalization has been emerged from a Logarithmic Sobolev Inequality approach. This introduced family of distributions result normal, uniform and Laplace distributions for certain values of the extra parameter. The moments of this distribution family are evaluated and extensively studied. Their behaviour, with respect to the involved extra parameter, is also examined and discussed,  as far as the variance and kurtosis are concerned. This extra parameter influences the shape of the members of the family and, therefore, “heavy-tailed” distributions can be obtained.