Results of studies that are based on the entire population cannot be applied on only a particular class of the society that possesses specific characteristics. The results of such studies will be inaccurate because of the negative influence of certain elements of society on the study group or non-importance of some elements of the society that can be neglected. Some conditions can be placed on the population so that the non-important part of the society can be trimmed if that part is outside the interest of the researcher. However, placing such conditions on the community, the data would not follow the same distribution of entire community. Hence, we need to find the distributional characteristics of the truncated data including the probability density functions and estimation of the parameters for the truncated distributions. This can be done for specific class distributions. In this thesis we discuss the distributional properties of and make statistical inference on the parameters of some truncated distributions among the exponential family such as the normal distribution, the Gamma distribution, and the Geometric distribution. Such probability distributions have different applications especially in economics and environmental applications. Therefore, we find the distributional properties and the estimates of the parameters of the normal distribution, the Gamma distribution, and the Geometric distribution. Finally, simulation studies were conducted to check the results obtained in this thesis.