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We consider the problem of energy efficient random deployment of sensor network. Our goal is to find the sensor node density, or alternatively, the energy resource density at every point inside a given deployment region, which results in allocating the minimum total number of deployed sensors, or alternatively, the minimum total energy source subject to constraints on the quality of monitoring (QoM) and network lifetime. The QoM is defined as the average of spatial distortion in reconstructed signal at the base station and can be bounded for a random deployment of sensor nodes when sensors are points of a Poisson process in the deployment region. To solve the optimization problem, we first determine a node density which satisfies the QoM constraint. Next we present a continuous space model for random deployment with the associated routing scheme that can be used to provide the minimum total required energy consumption. Finally, we present a spatial distribution of the sensor nodes (or the energy resources) that can achieve this minimum total energy. Simulation result shows that the minimum total energy obtained is close to the actual energy required in a randomly deployed dense network.