"Scaled largest eigenvalue (SLE) detector stands out as the best single-primary-user detector in uncertain noisy environments. In this paper, we consider a multi-antenna cognitive radio system in which we aim at detecting the presence/absence of a primary user (PU) using the SLE detector. We study the distribution of the SLE as a large num- ber of samples are used in detection without constraint on the number of antennas. By the exploitation of the distributions of the largest eigen- value and the trace of the receiver sample covariance matrix, we show that the SLE could be modeled as a normal random variable. Moreover, we derive the distribution of the SLE and deduce a simple yet accurate form of the probability of false alarm. Hence, this derivation yields a very simple form of the detection threshold. The analytical derivations are validated through extensive Monte Carlo simulations."