So far, the lack of a common, effective way of measuring energy poverty has been detected as a major weakness in handling the energy poverty problem. One of the main causes has been the complexity of modeling the "required energy consumption" of households, as demanded by the official 10% indicator, and its replacement in calculations by the "actual energy consumption", which, as is well known, underestimates the real needs of households. This weakness is addressed in this paper, through the development of the "Stochastic Model of Energy Poverty" (SMEP). The development of the model includes, firstly, the modeling of energy consumption at household level and, subsequently, the transition from household level to country level through stochastic analysis (Monte-Carlo simulation). Through Sensitivity Analysis, the impact of various parameters on energy poverty is quantified for the first time, by determining their weighting factors. Applied to the case of Greece, it is found that energy poverty reaches 70.4%, with income being the decisive factor affecting energy poverty at 63%, while other variables (Hint, etc.) follow at significantly lower percentages. The findings can be used in order to assess in advance the effectiveness of energy poverty measures, making the model a valuable policy tool.