In a renewal process, the interarrival times are independent and identically distributed, and in a renewal reward process, the pairs of reward and interarrival time are i.i.d. Many useful results hold for these processes. This paper relaxes the assumption of identical distributions while keeping the assumption of independence. This paper explores the properties of the mean average number of occurrence of events and the mean average reward on any finite time interval. The paper also discusses the limiting properties of these two quantities and extends many results from the renewal process and renewal reward process to the more general counting process and reward sequence.