  The basic properties of nonlinear ion-acoustic (IA) waves (IAWs),
particularly finite amplitude IA rogue waves (IARWs) in a plasma medium
(containing pair ions, iso-thermal positrons and non-thermal electrons) are
theoretically investigated by deriving the nonlinear Schr\"{o}dinger equation
(NLSE). The criteria for the modulational instability of IAWs, and the basic
features of finite amplitude IARWs are identified. The modulationally stable
and unstable regions are determined by the sign of the ratio of the dispersive
coefficient to the nonlinear coefficient of NLSE. The latter is analyzed to
obtain the region for the existence of the IARWs, which corresponds to the
unstable region. The shape of the profile of the rogue waves depends on the
non-thermal parameter $\alpha$ and the ratio of electron temperature to
positron temperature. It is found that the increase in the value of the
non-thermal parameter enhances both the amplitude and width of IARWs, and that
the enhancement of positron (electron) temperature reduces (enhances) the
amplitude and width of IARWs. It is worth to mention that our present
investigation may be useful for understanding the salient features of IARWs in
space (viz., upper region of Titan's atmosphere, cometary comae, and Earth's
ionosphere, etc.) and laboratory (viz., plasma processing reactor and neutral
beam sources, etc.) plasmas.
