In reply to KAUSHIK BHATTACHARJEE.

I mean you can translate and scale the distributions to have mean=mu and var=sigma^2. Since they are symmetric, they have skew=0.

The history of this problem goes back to Pearson in the 1800s, with other contributions by Burr and N. Johnson. Each of these researchers developed a system of distributions that depend on a small number of parameters (four or less) that can achieve any skewness and kurtosis that is feasible. In modern times, Leemis and others have overlayed classical distributions (normal, chi-square, gamma, etc) on one diagram that shows how various distributions share the same skewness and kurtosis even though they are different distributions (higher-order moments are different). The graphical picture that unifies this research is called the moment-ratio diagram. See "The Moment-Ratio Diagram" for more information.