The gamma distribution is highly important in applications and data modeling. It is usually used to model waiting times, the size of insurance claims, and rainfalls. In this paper we state and prove two new characterizations of the two parameter gamma distribution by establishing a connection between s-right truncated moments (s-left truncated moments) and the reversed hazard rate (hazard rate). These characterization results are easier to check in data analysis. Besides, our results generalize some of the well-known theoretical results of Koicheva (1993) and Ahsanullah et al., 2012. Keywords: characterization; Failure rate; Reversed failure rate; Right censored mean function; Left censored mean function; Gamma distribution.