Abstract: We combine two well-known statements of results in the statistical and mathematical literature, one related to symmetric continuous distributions and the other to the integration of functions, to obtain some new results regarding symmetric distributions and involving the value at risk and the tail value at risk, well-known tools used in actuarial and financial statistics, among others. Generalizations of the skew normal distribution in its univariate and multivariate versions obtained from one of the results are also shown, and a new method is proposed for generating families of skewed continuous distributions.

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