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We consider a downlink wireless system with a multi-antenna base station (BS) and single-antenna users. The error in the channel knowledge at the BS is assumed to have Gaussian distribution. Power allocation strategies are designed in order to satisfy the users' quality-of-service targets with certain probabilities. Conservative solutions of the problems are found by applying the Vysochanskii-Petunin inequality in combination with the theory of interference functions. Significant performance improvements are obtained comparing with methods based on the worst-case optimization.