Economic production quantity models usually assume that the production is fixed and finite. However, due to various random factors effecting the production, the production process becomes random. This paper deals with the development and analysis of economic production quantity model in which the production is random and follows a generalized Pareto distribution. The generalized Pareto distribution is capable of including different types of production rates. Here it is further assumed that the lifetime of the commodity is random and follows a two parameter Weibull distribution. The Weibull decay includes constant, increasing and decreasing rates of deterioration. It is also assumed that the demand is dependent on selling price. Assuming that shortages are allowed and fully backlogged the instantaneous state of on hand inventory is derived. With suitable cost considerations the total cost function and profit rate function are obtained and minimized with respect to the production uptime and downtime. The optimal production uptime, downtime, production quantity and selling price are derived. A numerical illustration demonstrating the solution procedure of the model is presented. The sensitivity analysis of the model revealed that the production and deteriorating distributions parameters have significant influence on the optimal production schedule and production quantity. This model is extended to the case of without shortages. This model also includes some of the earlier models as particular cases for specific or limiting values of the parameters.