Some counterexamples of a skew-normal distribution

Counterexamples of a skew-normal distribution are developed to improve our understanding of this distribution. Two examples on bivariate non-skew-normal distribution owning marginal skew-normal distributions are first provided. Sum of dependent skew-normal and normal variables does not follow a skew-normal distribution. Continuous bivariate density with discontinuous marginal density also exists in skew-normal distribution. An example presents that the range of possible correlations for bivariate skew-normal distribution is constrained in a relatively small set. For unified skew-normal variables, an example about converging in law are discussed. Convergence in distribution is involved in two separate examples for skew-normal variables. The point estimation problem, which is not a counterexample, is provided because of its importance in understanding the skew-normal distribution. These materials are useful for undergraduate and/or graduate teaching courses.