The first negative moment in the sense of the Cauchy principal value

Alternative sufficient conditions, for the existence and finiteness of the first negative moment of a continuous density function defined on the real line, are established in the sense of the (Cauchy) principal value. Using the principal value sense for practical applications, we derive explicit expressions of the reciprocal moment of a skew-normal distribution. In addition, some interesting properties discovered from the skew-symmetric model are also obtained. Finally, a case example is used to illustrate the concepts in lifetime data analysis.