This paper aims to construct a framework of evidence theory by normal possibility distributions defined by exponential functions. Since possibility distributions are obtained by an expert knowledge or can be identified by given data, a possibility distribution is regarded as an evidence in this paper. A rule of combination of evidences is given with the same concept as. Dempster's rule. Also, measures of ignorance and fuzziness of an evidence are defined by a normality factor and the area of a possibility distribution, respectively. These definitions are similar to those given by G. Shafer. Next, marginal and conditional possibilities are defined from a joint possibility distribution and it is shown that these definitions are well matched to each other. Thus, the posterior possibility is derived from the prior possibility in the same form as Bayes' formula. This shows the possibility that an information-decision theory can be reconstructed from the viewpoint of possibility distributions. Furthermore, linear systems whose variables are defined by possibility distributions are discussed. Operations of fuzzy vectors by multi-dimensional possibility distributions are well formulated, using the extension principle of L. A. Zadeh. Last, some comment on an application of possibility distributions is given in a discriminant analysis using fuzzy if-then rules.