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Optimal power scheduling for distributed detection in a Gaussian sensor network is addressed for both independent and correlated observations. We assume amplify-and-forward local processing at each node. The wireless link between sensors and the fusion center is assumed to undergo fading and coefficients are assumed to be available at the transmitting sensors. The objective is to minimize the total network power to achieve a desired fusion error probability at the fusion center. For i.i.d. observations, the optimal power allocation is derived analytically in closed form. When observations are correlated, first, an easy to optimize upper bound is derived for sufficiently small correlations and the power allocation scheme is derived accordingly. Next, an evolutionary computation technique based on particle swarm optimization is developed to find the optimal power allocation for arbitrary correlations. The optimal power scheduling scheme suggests that the sensors with poor observation quality and bad channels should be inactive to save the total power expenditure of the system. It is shown that the probability of fusion error performance based on the optimal power allocation scheme outperforms the uniform power allocation scheme especially when either the number of sensors is large or the local observation quality is good.