Coprime sensor arrays (CSAs) achieve the resolution of a fully populated uniform linear array (ULA) with the same aperture using fewer sensors. The CSA's reduced number of sensors diminishes its ability to attenuate white noise, and consequently also reduces its array gain. Assuming narrowband independent signal and white noise, when the signal and the noise are modeled by circular complex zero mean Gaussian distributions, the detection statistic distribution for the ULA simplifies to an exponential distribution with its parameter dependent on the input SNR. The detection statistic for the CSA is the product of the output of one subarray with the complex conjugate of the output of the second subarray. Being the product of two independent complex Gaussian random variables, the CSA detection statistic has a distribution proportional to a modified zeroth order Bessel function of the second kind [O'Donoughue and Moura, IEEE Signal Process. (2012)]. Manipulating the ULA and CSA detection statistic distributions provides analytical expressions for probabilities of detection and false alarm and also mean discriminating information. Evaluating these expressions confirms that a CSA pays a detection gain penalty of about 5 dB over a wide range of SNRs and array sizes. [Work supported by ONR.]