This book provides expository derivations for moments of a family of pseudo distributions, which is an extended family of distributions including the pseudo normal (PN) distributions recently proposed by the author. The PN includes the skew normal (SN) derived by A. Azzalini and the closed skew normal (CSN) obtained by A. Domínguez-Molina, G. González-Farías, and A. K. Gupta as special cases. It is known that the CSN includes the SN and other various distributions as special cases, which shows that the PN has a wider variety of distributions. The SN and CSN have symmetric and skewed asymmetric distributions. However, symmetric distributions are restricted to normal ones. On the other hand, symmetric distributions in the PN can be non-normal as well as normal. In this book, for the non-normal symmetric distributions, the term ¿kurtic normal (KN)¿ is used, where the coined word ¿kurtic¿ indicates ¿mesokurtic, leptokurtic, or platykurtic¿ used in statistics. The variety of the PN was made possible using stripe (tigerish) and sectional truncation in univariate and multivariate distributions, respectively. The proofs of the moments and associated results are not omitted and are often given in more than one method with their didactic explanations.

Haruhiko Ogasawara is Professor Emeritus, Otaru University of Commerce.

The Sectionally Truncated Normal Distribution.- Normal Moments Under Stripe Truncation and the Real-Valued Poisson Distribution.- The Basic Parabolic Cylinder Distribution and its Multivariate Extension.- The Pseudo-Normal (PN) Distribution.- The Kurtic-Normal (KN) Distribution.- The Normal-Normal (NN) Distribution.- The Decompositions of the PN and NN Distributed Variables.- The Truncated Pseudo-Normal (TPN) and Truncated Normal-Normal (TNN) Distributions.- The Student t- and Pseudo-t (PT) Distributions: Various Expressions of Mixtures.- Multivariate Measures of Skewness and Kurtosis.