Assessment of disclosure risk is of paramount importance in data privacy research and applications. The concept of differential privacy (DP) formalizes privacy in probabilistic terms and provides a robust concept for privacy protection. Practical applications of DP involve development of DP mechanisms to release data at a pre-specified privacy budget. In this paper, we generalize the widely used Laplace mechanism to the family of generalized Gaussian (GG) mechanism based on the <inline-formula><tex-math notation="LaTeX">$l_p$</tex-math><alternatives><inline-graphic xlink:href="liu-ieq1-2845388.gif"/></alternatives></inline-formula> global sensitivity of statistical queries. We explore the theoretical requirement for the GG mechanism to reach DP at prespecified privacy parameters, and investigate the connections and differences between the GG mechanism and the Exponential mechanism based on the GG distribution. We also present a lower bound on the scale parameter of the Gaussian mechanism of <inline-formula><tex-math notation="LaTeX">$(\epsilon, \delta)$</tex-math><alternatives><inline-graphic xlink:href="liu-ieq2-2845388.gif"/></alternatives></inline-formula>-probabilistic DP as a special case of the GG mechanism, and compare the utility of sanitized results in the tail probability and dispersion between the Gaussian and Laplace mechanisms. Lastly, we apply the GG mechanism in three experiments and compare the accuracy of sanitized results in the <inline-formula><tex-math notation="LaTeX">$l_1$</tex-math><alternatives><inline-graphic xlink:href="liu-ieq3-2845388.gif"/></alternatives></inline-formula> distance and Kullback-Leibler divergence, and examine the prediction power of a SVM classifier constructed with the sanitized data relative to the original results.