Many families of density functions approach the normal one as a certain parameter tends to an appropriate value . However , there are only a few parametric 0cm 0cm 0pt; LINE-HEIGHT: 115%; TEXT-ALIGN: justify" It would be ideal to have at hand a 0cm 0cm 0pt; LINE-HEIGHT: 115%; TEXT-ALIGN: justify" (i) " strict inclusion " of the normal density, (ii) mathematical tractability, (iii) wide range of the indices of skewness and kurtosis . The term skew-normal (SN) refers to a parametric stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" writte , if its density function is where and denote the N(0,1) density and distribution function , respectively; the parameter which regulates the skewness varies in and corresponds to the N(0,1) density . In past two decades , the type="#_x0000_t75" the noncentral skew chi-square distribution is defined and the properties of quadratic forms are studied in this paper . In the last part of this thesis , the necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained by using their moment generating functions .