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In this paper, a new mixture family of multivariate normal distributions, formed by mixing multivariate normal distribution and skewed distribution, is constructed. Some properties of this family, such as characteristic function, moment generating function, and the first four moments are derived. The distributions of affine transformations and canonical forms of the model are also derived. An EM type algorithm is developed for the maximum likelihood estimation of model parameters. We have considered in detail, some special cases of the family, using standard gamma and standard exponential mixture distributions, denoted by MMNG and MMNE, respectively. For the proposed family of distributions, different multivariate measures of skewness are computed. In order to examine the performance of the developed estimation method, some simulation studies are carried out to show that the maximum likelihood estimates based on the EM type algorithm do provide good performance. For different choices of parameters of MMNE distribution, several multivariate measures of skewness are computed and compared. Because some measures of skewness are scalar and some are vectors, in order to evaluate them properly, we have carried out a simulation study to determine the power of tests, based on sample versions of skewness measures as test statistics to test the fit of the MMNE distribution. Finally, two real data sets are used to illustrate the usefulness of the proposed family of distributions and the associated inferential method.