We consider the problem of detecting a sparse random signal from the compressive measurements without reconstructing the signal. Using a subspace model for the sparse signal where the signal parameters are drawn according to Gaussian law, we obtain the detector based on Neyman-Pearson criterion and analytically determine its operating characteristics when the signal covariance is known. These results are extended to situations where the covariance cannot be estimated. The results can be used to determine the number of measurements needed for a particular detector performance and also illustrate the presence of an optimal support for a given number of measurements.